Shortest Path<\/strong><\/td>The path between two vertices that has the minimum sum of edge weights.<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
\n\n\n\nWhat is Dijkstra’s Algorithm?<\/h2>\n\n\n\nDijkstra’s Algorithm is the classic greedy algorithm designed to solve the single-source shortest path<\/strong> problem on a weighted graph.<\/p>\n\n\n\nProblem Statement:<\/h3>\n\n\n\n\n- Given:<\/strong> A weighted graph with non-negative edge weights and a starting source vertex (s).<\/li>\n\n\n\n
- Find:<\/strong> The shortest path from (s) to every other reachable vertex in the graph.<\/li>\n\n\n\n
- Output:<\/strong><\/li>\n\n\n\n
dist[]<\/code>: An array where dist[v]<\/code> is the shortest distance from source (s) to vertex (v).<\/li>\n\n\n\nparent[]<\/code>: An array where parent[v]<\/code> holds the predecessor of (v) on the shortest path (used for path reconstruction).<\/li>\n<\/ul>\n\n\n\nReal-world Analogy (GPS Navigation):<\/h3>\n\n\n\nImagine you are at home, and Google Maps needs to find the fastest route to a restaurant. Google’s servers run Dijkstra’s algorithm on a road network graph where each road’s weight is the travel time. In a single run, Dijkstra computes the fastest path not just to your chosen restaurant, but to every<\/strong> reachable location from your house. This is why Google Maps can instantly show you alternative routes and arrival times to nearby places\u201a\u00c4\u00eethey are all computed at once!<\/p>\n\n\n\nIntuition (The Water Flood Analogy):<\/h3>\n\n\n\nImagine water flooding a network of pipes from your starting location. The water spreads in all directions simultaneously but travels faster through shorter, lower-resistance paths. The first moment water reaches a city is guaranteed to be the fastest path. Dijkstra’s algorithm simulates this process step-by-step.<\/p>\n\n\n\n
\n\n\n\nStep-by-Step Algorithm<\/h2>\n\n\n\n1. Initialization:<\/h3>\n\n\n\n\n- Set
dist[source] = 0<\/code> and dist[v] = \u221e;<\/code> for all other vertices (v).<\/li>\n\n\n\n- Set
parent[v] = null<\/code> for all vertices.<\/li>\n\n\n\n- Create an empty priority queue (min-heap)
pq<\/code> and push (0, source)<\/code> into it.<\/li>\n\n\n\n- Keep track of visited nodes using an empty set or array.<\/li>\n<\/ol>\n\n\n\n
2. Main Loop:<\/h3>\n\n\n\nWhile the priority queue pq<\/code> is not empty:<\/p>\n\n\n\n\n- Extract the pair
(distance, u)<\/code> with the minimum distance.<\/li>\n\n\n\n- If
u<\/code> is already visited, skip it.<\/li>\n\n\n\n- Mark
u<\/code> as visited.<\/li>\n\n\n\n- For each neighbor
v<\/code> of u<\/code>:<\/li>\n<\/ol>\n\n\n\n\n- Calculate
new_distance = dist[u] + weight(u, v)<\/code>.<\/li>\n\n\n\n- Relaxation Step:<\/strong> If
new_distance < dist[v]<\/code>:\n\n- Update
dist[v] = new_distance<\/code>.<\/li>\n\n\n\n- Update
parent[v] = u<\/code>.<\/li>\n\n\n\n- Push
(new_distance, v)<\/code> into the priority queue.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n
\n\n\n\nPseudocode<\/h2>\n\n\n\nAlgorithm: Dijkstra(graph, source)\nInput: A graph with vertices and weighted edges, and a starting vertex source\nOutput: dist[] (shortest distances) and parent[] (predecessor path trackers)\n\nfor each vertex v in graph:\n dist[v] = INFINITY\n parent[v] = NULL\n\ndist[source] = 0\npriority_queue pq = empty min-heap\npq.push((0, source))\nvisited = empty set\n\nwhile pq is not empty:\n current_dist, u = pq.pop() # Extract the vertex with the minimum distance\n\n if u in visited:\n continue # Skip already processed vertices\n\n visited.add(u)\n\n for each vertex v adjacent to u:\n weight = edge_weight(u, v)\n\n # Relaxation Step\n if dist[u] + weight < dist[v]:\n dist[v] = dist[u] + weight\n parent[v] = u\n pq.push((dist[v], v))\n\nreturn dist, parent<\/code><\/pre>\n\n\n\n
\n\n\n\nWalkthrough Example<\/h2>\n\n\n\nGraph Definition:<\/h3>\n\n\n\n\n- Vertices:<\/strong> A, B, C, D, E<\/li>\n\n\n\n
- Edges:<\/strong> A-B (4), A-C (2), B-C (1), B-D (5), B-E (6), C-D (8), C-E (10), D-E (2)<\/li>\n<\/ul>\n\n\n\n
Step 1: Initialization<\/h4>\n\n\n\ndist: [A:0, B:\u221e, C:\u221e, D:\u221e, E:\u221e]\nparent: [A:null, B:null, C:null, D:null, E:null]\nvisited: []\npq: [(0, A)]<\/code><\/pre>\n\n\n\nStep 2: Process A<\/h4>\n\n\n\n\n- Extract
(0, A)<\/code> from pq<\/code> and mark A<\/code> as visited.<\/li>\n\n\n\n- Explore
A<\/code>‘s neighbors:<\/li>\n\n\n\n- B:<\/strong>
0 + 4 = 4 <<\/code> \u221e<\/code>. Update dist[B] = 4<\/code>, parent[B] = A<\/code>.<\/li>\n\n\n\n- C:<\/strong>
0 + 2 = 2 < \u221e<\/code>. Update dist[C] = 2<\/code>, parent[C] = A<\/code>.<\/li>\n<\/ul>\n\n\n\ndist: [A:0, B:4, C:2, D:∞, E:∞]\nvisited: [A]\npq: [(2, C), (4, B)]<\/code><\/pre>\n\n\n\nStep 3: Process C<\/h4>\n\n\n\n\n- Extract
(2, C)<\/code> from pq<\/code> and mark C<\/code> as visited.<\/li>\n\n\n\n- Explore
C<\/code>‘s neighbors:<\/li>\n\n\n\n- A:<\/strong> Already visited, skip.<\/li>\n\n\n\n
- B:<\/strong>
2 + 1 = 3 < 4<\/code>. Update dist[B] = 3<\/code>, parent[B] = C<\/code>.<\/li>\n\n\n\n- D:<\/strong>
2 + 8 = 10 < \u221e<\/code>. Update dist[D] = 10<\/code>, parent[D] = C<\/code>.<\/li>\n\n\n\n- E:<\/strong>
2 + 10 = 12 < \u221e<\/code>. Update dist[E] = 12<\/code>, parent[E] = C<\/code>.<\/li>\n<\/ul>\n\n\n\ndist: [A:0, B:3, C:2, D:10, E:12]\nvisited: [A, C]\npq: [(3, B), (4, B), (10, D), (12, E)]<\/code><\/pre>\n\n\n\nStep 4: Process B<\/h4>\n\n\n\n\n- Extract
(3, B)<\/code> from pq<\/code> and mark B<\/code> as visited.<\/li>\n\n\n\n- Explore
B<\/code>‘s neighbors:<\/li>\n\n\n\n- A, C:<\/strong> Already visited, skip.<\/li>\n\n\n\n
- D:<\/strong>
3 + 5 = 8 < 10<\/code>. Update dist[D] = 8<\/code>, parent[D] = B<\/code>.<\/li>\n\n\n\n- E:<\/strong>
3 + 6 = 9 < 12<\/code>. Update dist[E] = 9<\/code>, parent[E] = B<\/code>.<\/li>\n<\/ul>\n\n\n\ndist: [A:0, B:3, C:2, D:8, E:9]\nvisited: [A, C, B]\npq: [(4, B), (8, D), (9, E), (10, D), (12, E)]<\/code><\/pre>\n\n\n\nStep 5: Process D<\/h4>\n\n\n\n\n- Extract
(4, B)<\/code> from pq<\/code>. Since B<\/code> is already visited, skip.<\/li>\n\n\n\n- Extract
(8, D)<\/code> from pq<\/code> and mark D<\/code> as visited.<\/li>\n\n\n\n- Explore
D<\/code>‘s neighbors:<\/li>\n\n\n\n- B, C:<\/strong> Already visited, skip.<\/li>\n\n\n\n
- E:<\/strong>
8 + 2 = 10 > 9<\/code>. No update needed (9 is already shorter).<\/li>\n<\/ul>\n\n\n\ndist: [A:0, B:3, C:2, D:8, E:9]\nvisited: [A, C, B, D]\npq: [(9, E), (10, D), (12, E)]<\/code><\/pre>\n\n\n\nStep 6: Process E<\/h4>\n\n\n\n\n- Extract
(9, E)<\/code> from pq<\/code> and mark E<\/code> as visited.<\/li>\n\n\n\nE<\/code> has no unvisited neighbors.<\/li>\n<\/ul>\n\n\n\ndist: [A:0, B:3, C:2, D:8, E:9]\nvisited: [A, C, B, D, E]\npq: empty<\/code><\/pre>\n\n\n\nFinal Results (Shortest Path from A):<\/h3>\n\n\n\n\n- A \u2192 A:<\/strong> Cost:
0<\/code><\/li>\n\n\n\n- A \u2192 B:<\/strong> Cost:
3<\/code> (Path: A \u2192 C \u2192 B<\/code>)<\/li>\n\n\n\n- A \u2192 C:<\/strong> Cost:
2<\/code> (Path: A \u2192 C<\/code>)<\/li>\n\n\n\n- A \u2192 D:<\/strong> Cost:
8<\/code> (Path: A \u2192 C \u2192 B \u2192 D<\/code>)<\/li>\n\n\n\n- A \u2192 E:<\/strong> Cost:
9<\/code> (Path: A \u2192 C \u2192 B \u2192 E<\/code>)<\/li>\n\n\n\n- Path Reconstruction to E:<\/strong>
E \u2190 B \u2190 C \u2190 A<\/code> (Reverse order of parents).<\/li>\n<\/ul>\n\n\n\n
\n\n\n\nComplexity Analysis<\/h2>\n\n\n\nTime Complexity:<\/h3>\n\n\n\n\n- With Binary Min-Heap:<\/strong> (O((V + E) log V))<\/strong><\/li>\n\n\n\n
- (V) insertions\/extractions from the heap: (O(V log V))<\/li>\n\n\n\n
- (E) relaxation operations (decrease-key): (O(E log V))<\/li>\n\n\n\n
- With Array (Linear Search):<\/strong> (O(V^2))<\/strong><\/li>\n\n\n\n
- Preferred for extremely dense graphs where (E \\approx V^2).<\/li>\n\n\n\n
- With Fibonacci Heap:<\/strong> (O(E + V log V))<\/strong><\/li>\n\n\n\n
- High constant factor; primarily of theoretical interest and rarely used in production.<\/li>\n<\/ul>\n\n\n\n
Space Complexity:<\/h3>\n\n\n\n\n- (O(V + E))<\/strong> to store the graph adjacency list representation.<\/li>\n\n\n\n
- (O(V))<\/strong> helper storage for
dist[]<\/code>, parent[]<\/code> tracking, and the priority queue states.<\/li>\n<\/ul>\n\n\n\n
\n\n\n\n\n\nInteractive Dijkstra’s Algorithm Visualizer<\/title>\n<link rel=\"preconnect\" href=\"https:\/\/fonts.googleapis.com\">\n<link rel=\"preconnect\" href=\"https:\/\/fonts.gstatic.com\" crossorigin=\"\">\n<link href=\"https:\/\/fonts.googleapis.com\/css2?family=Outfit:wght@300;400;600;800&family=Plus+Jakarta+Sans:wght@300;400;500;600;700&display=swap\" rel=\"stylesheet\">\n\n<style>\n #dijkstra-widget {\n --bg-dark: #0f172a;\n --bg-card: rgba(30, 41, 59, 0.7);\n --border-color: rgba(255, 255, 255, 0.08);\n --text-main: #f8fafc;\n --text-muted: #94a3b8;\n \n \/* Theme colors *\/\n --color-primary: #3b82f6; \/* Blue for general *\/\n --color-success: #10b981; \/* Green for visited *\/\n --color-warning: #f59e0b; \/* Amber for currently processing *\/\n --color-danger: #ef4444;\n --color-accent: #a855f7; \/* Purple for queue\/updating *\/\n \n \/* Neon glow effects *\/\n --glow-visited: 0 0 15px rgba(16, 185, 129, 0.6);\n --glow-processing: 0 0 15px rgba(245, 158, 11, 0.8);\n --glow-updating: 0 0 15px rgba(168, 85, 247, 0.6);\n\n \/* Scoped body styles *\/\n font-family: 'Plus Jakarta Sans', sans-serif;\n background-color: var(--bg-dark);\n background-image: \n radial-gradient(at 0% 0%, rgba(59, 130, 246, 0.1) 0px, transparent 50%),\n radial-gradient(at 100% 100%, rgba(168, 85, 247, 0.1) 0px, transparent 50%);\n color: var(--text-main);\n width: 100%;\n max-width: 1200px;\n margin: 1.5rem auto;\n padding: 2rem 1.5rem;\n border-radius: 16px;\n box-shadow: 0 12px 40px rgba(0, 0, 0, 0.6);\n border: 1px solid var(--border-color);\n transition: background 0.3s, border-color 0.3s, box-shadow 0.3s;\n text-align: left;\n }\n\n #dijkstra-widget * {\n box-sizing: border-box;\n margin: 0;\n padding: 0;\n }\n\n #dijkstra-widget header {\n text-align: center;\n margin-bottom: 2rem;\n }\n\n #dijkstra-widget h1 {\n font-family: 'Outfit', sans-serif;\n font-size: 2.5rem;\n font-weight: 800;\n background: linear-gradient(135deg, #60a5fa 0%, #c084fc 100%);\n -webkit-background-clip: text;\n -webkit-text-fill-color: transparent;\n margin-bottom: 0.5rem;\n letter-spacing: -0.02em;\n }\n\n #dijkstra-widget header p {\n color: var(--text-muted);\n font-size: 1.1rem;\n }\n\n \/* Grid Layout *\/\n #dijkstra-widget .visualizer-grid {\n display: grid;\n grid-template-columns: 1.2fr 0.8fr;\n gap: 1.5rem;\n margin-bottom: 1.5rem;\n }\n\n @media (max-width: 968px) {\n #dijkstra-widget .visualizer-grid {\n grid-template-columns: 1fr;\n }\n }\n\n #dijkstra-widget .card {\n background: var(--bg-card);\n backdrop-filter: blur(12px);\n -webkit-backdrop-filter: blur(12px);\n border: 1px solid var(--border-color);\n border-radius: 16px;\n padding: 1.5rem;\n box-shadow: 0 10px 30px -10px rgba(0,0,0,0.5);\n display: flex;\n flex-direction: column;\n position: relative;\n overflow: hidden;\n }\n\n #dijkstra-widget .card-title {\n font-family: 'Outfit', sans-serif;\n font-size: 1.25rem;\n font-weight: 600;\n margin-bottom: 1rem;\n display: flex;\n align-items: center;\n gap: 0.5rem;\n color: var(--text-main);\n border-bottom: 1px solid var(--border-color);\n padding-bottom: 0.5rem;\n }\n\n \/* Graph Section *\/\n #dijkstra-widget .graph-container {\n display: flex;\n justify-content: center;\n align-items: center;\n background: rgba(15, 23, 42, 0.5);\n border-radius: 12px;\n border: 1px solid rgba(255, 255, 255, 0.04);\n padding: 1rem;\n position: relative;\n height: 380px;\n }\n\n #dijkstra-widget #graph-svg {\n width: 100%;\n height: 100%;\n }\n\n #dijkstra-widget svg {\n user-select: none;\n }\n\n \/* SVG Elements styling *\/\n #dijkstra-widget .edge {\n stroke: #475569;\n stroke-width: 3;\n stroke-linecap: round;\n transition: stroke 0.3s, stroke-width 0.3s, stroke-dasharray 0.3s;\n }\n\n #dijkstra-widget .edge.active {\n stroke: var(--color-success);\n stroke-width: 4;\n }\n\n #dijkstra-widget .edge.updating {\n stroke: var(--color-accent);\n stroke-width: 4;\n stroke-dasharray: 8 4;\n animation: dash 1s linear infinite;\n }\n \n #dijkstra-widget .edge.shortest-path {\n stroke: #38bdf8;\n stroke-width: 5;\n }\n\n @keyframes dash {\n to {\n stroke-dashoffset: -12;\n }\n }\n\n #dijkstra-widget .edge-label-bg {\n fill: var(--bg-dark);\n rx: 4;\n ry: 4;\n }\n\n #dijkstra-widget .edge-label {\n fill: var(--text-muted);\n font-size: 14px;\n font-weight: 600;\n text-anchor: middle;\n dominant-baseline: central;\n }\n\n #dijkstra-widget .edge-label.active {\n fill: var(--text-main);\n }\n\n #dijkstra-widget .node-circle {\n fill: #1e293b;\n stroke: #64748b;\n stroke-width: 3;\n transition: fill 0.3s, stroke 0.3s, filter 0.3s;\n cursor: pointer;\n }\n\n #dijkstra-widget .node-circle:hover {\n fill: #334155;\n }\n\n #dijkstra-widget .node.visited .node-circle {\n fill: rgba(16, 185, 129, 0.15);\n stroke: var(--color-success);\n filter: drop-shadow(var(--glow-visited));\n }\n\n #dijkstra-widget .node.processing .node-circle {\n fill: rgba(245, 158, 11, 0.2);\n stroke: var(--color-warning);\n filter: drop-shadow(var(--glow-processing));\n animation: pulse 2s infinite;\n }\n\n #dijkstra-widget .node.target .node-circle {\n fill: rgba(168, 85, 247, 0.2);\n stroke: var(--color-accent);\n filter: drop-shadow(var(--glow-updating));\n }\n\n @keyframes pulse {\n 0% {\n stroke-width: 3px;\n }\n 50% {\n stroke-width: 5px;\n }\n 100% {\n stroke-width: 3px;\n }\n }\n\n #dijkstra-widget .node-text {\n fill: var(--text-main);\n font-size: 16px;\n font-weight: 700;\n text-anchor: middle;\n dominant-baseline: central;\n pointer-events: none;\n }\n\n \/* Control Panel *\/\n #dijkstra-widget .controls-card {\n width: 100%;\n margin-bottom: 1.5rem;\n }\n\n #dijkstra-widget .controls-wrapper {\n display: flex;\n flex-wrap: wrap;\n gap: 1rem;\n align-items: center;\n justify-content: space-between;\n }\n\n #dijkstra-widget .btn-group {\n display: flex;\n gap: 0.5rem;\n }\n\n #dijkstra-widget button {\n background: rgba(255, 255, 255, 0.05);\n border: 1px solid var(--border-color);\n color: var(--text-main);\n padding: 0.6rem 1.2rem;\n border-radius: 8px;\n cursor: pointer;\n font-weight: 600;\n font-size: 0.9rem;\n display: flex;\n align-items: center;\n gap: 0.5rem;\n transition: background 0.2s, border-color 0.2s, transform 0.1s;\n }\n\n #dijkstra-widget button:hover:not(:disabled) {\n background: rgba(255, 255, 255, 0.1);\n border-color: rgba(255, 255, 255, 0.2);\n }\n\n #dijkstra-widget button:active:not(:disabled) {\n transform: scale(0.97);\n }\n\n #dijkstra-widget button:disabled {\n opacity: 0.4;\n cursor: not-allowed;\n }\n\n #dijkstra-widget button.primary {\n background: var(--color-primary);\n border-color: transparent;\n }\n\n #dijkstra-widget button.primary:hover:not(:disabled) {\n background: #2563eb;\n }\n\n #dijkstra-widget .slider-container {\n display: flex;\n align-items: center;\n gap: 1rem;\n flex-grow: 1;\n max-width: 400px;\n }\n\n #dijkstra-widget .slider-label {\n font-size: 0.9rem;\n color: var(--text-muted);\n white-space: nowrap;\n min-width: 80px;\n }\n\n #dijkstra-widget input[type=\"range\"] {\n -webkit-appearance: none;\n width: 100%;\n height: 6px;\n border-radius: 3px;\n background: #334155;\n outline: none;\n cursor: pointer;\n }\n\n #dijkstra-widget input[type=\"range\"]::-webkit-slider-thumb {\n -webkit-appearance: none;\n width: 18px;\n height: 18px;\n border-radius: 50%;\n background: var(--color-primary);\n box-shadow: 0 0 10px rgba(59, 130, 246, 0.5);\n transition: transform 0.1s;\n }\n\n #dijkstra-widget input[type=\"range\"]::-webkit-slider-thumb:hover {\n transform: scale(1.2);\n }\n\n \/* State & Info Panel Styling *\/\n #dijkstra-widget .step-desc {\n font-size: 0.95rem;\n line-height: 1.5;\n color: var(--text-main);\n background: rgba(15, 23, 42, 0.4);\n border-left: 4px solid var(--color-primary);\n padding: 0.75rem 1rem;\n border-radius: 0 8px 8px 0;\n margin-bottom: 1.25rem;\n min-height: 80px;\n }\n\n #dijkstra-widget .step-desc.visited-update {\n border-left-color: var(--color-success);\n }\n\n #dijkstra-widget .state-section {\n margin-bottom: 1.25rem;\n }\n\n #dijkstra-widget .state-section-title {\n font-size: 0.85rem;\n text-transform: uppercase;\n letter-spacing: 0.05em;\n color: var(--text-muted);\n margin-bottom: 0.5rem;\n font-weight: 700;\n }\n\n \/* Badges list *\/\n #dijkstra-widget .badge-list {\n display: flex;\n flex-wrap: wrap;\n gap: 0.5rem;\n }\n\n #dijkstra-widget .badge {\n display: inline-flex;\n align-items: center;\n padding: 0.25rem 0.6rem;\n border-radius: 6px;\n font-size: 0.85rem;\n font-weight: 600;\n background: rgba(255, 255, 255, 0.05);\n border: 1px solid var(--border-color);\n }\n\n #dijkstra-widget .badge.visited {\n background: rgba(16, 185, 129, 0.1);\n border-color: rgba(16, 185, 129, 0.2);\n color: #34d399;\n }\n\n #dijkstra-widget .badge.processing {\n background: rgba(245, 158, 11, 0.1);\n border-color: rgba(245, 158, 11, 0.2);\n color: #fbbf24;\n }\n \n #dijkstra-widget .badge.queue-item {\n background: rgba(168, 85, 247, 0.1);\n border-color: rgba(168, 85, 247, 0.2);\n color: #c084fc;\n }\n\n \/* Tables *\/\n #dijkstra-widget table {\n width: 100%;\n border-collapse: collapse;\n font-size: 0.9rem;\n text-align: left;\n }\n\n #dijkstra-widget th, #dijkstra-widget td {\n padding: 0.5rem 0.75rem;\n border-bottom: 1px solid rgba(255, 255, 255, 0.05);\n }\n\n #dijkstra-widget th {\n color: var(--text-muted);\n font-weight: 600;\n font-size: 0.8rem;\n text-transform: uppercase;\n }\n\n #dijkstra-widget tr:last-child td {\n border-bottom: none;\n }\n\n #dijkstra-widget .val-cell {\n font-weight: 700;\n }\n\n #dijkstra-widget .updated-val {\n animation: highlight-green 1.5s ease-out;\n }\n\n @keyframes highlight-green {\n 0% {\n color: var(--color-success);\n background: rgba(16, 185, 129, 0.25);\n border-radius: 4px;\n }\n 100% {\n color: inherit;\n background: transparent;\n }\n }\n \n #dijkstra-widget .final-path-box {\n background: rgba(56, 189, 248, 0.1);\n border: 1px solid rgba(56, 189, 248, 0.2);\n padding: 0.75rem;\n border-radius: 8px;\n font-family: monospace;\n font-size: 1rem;\n color: #38bdf8;\n display: flex;\n flex-direction: column;\n gap: 0.5rem;\n }\n\n \/* Embedded iframe optimization *\/\n #dijkstra-widget .embed-info {\n text-align: center;\n margin-top: 1.5rem;\n font-size: 0.85rem;\n color: var(--text-muted);\n }\n\n #dijkstra-widget .embed-code {\n background: #090d16;\n border: 1px solid rgba(255, 255, 255, 0.05);\n border-radius: 6px;\n padding: 0.5rem;\n font-family: monospace;\n font-size: 0.8rem;\n margin-top: 0.5rem;\n display: inline-block;\n max-width: 100%;\n overflow-x: auto;\n color: #e2e8f0;\n }\n\n \/* Arrow in list *\/\n #dijkstra-widget .arrow-icon {\n margin: 0 0.25rem;\n color: var(--text-muted);\n }\n\n \/* Info grid at bottom *\/\n #dijkstra-widget .info-grid {\n display: grid;\n grid-template-columns: 1fr 1fr;\n gap: 1.5rem;\n margin-top: 1.5rem;\n margin-bottom: 2rem;\n }\n\n @media (max-width: 768px) {\n #dijkstra-widget .info-grid {\n grid-template-columns: 1fr;\n }\n }\n\n #dijkstra-widget .info-list {\n list-style: none;\n display: flex;\n flex-direction: column;\n gap: 0.75rem;\n }\n\n #dijkstra-widget .info-list li {\n display: flex;\n align-items: flex-start;\n gap: 0.5rem;\n font-size: 0.95rem;\n line-height: 1.5;\n }\n\n #dijkstra-widget .info-list li span.icon {\n flex-shrink: 0;\n margin-top: 0.1rem;\n }\n\n #dijkstra-widget .info-card-success {\n border-top: 4px solid var(--color-success) !important;\n }\n\n #dijkstra-widget .info-card-warning {\n border-top: 4px solid var(--color-danger) !important;\n }\n\n #dijkstra-widget .alert-box {\n background: rgba(239, 68, 68, 0.08);\n border: 1px solid rgba(239, 68, 68, 0.2);\n border-radius: 8px;\n padding: 1rem;\n margin-top: 1rem;\n }\n\n #dijkstra-widget .alert-box-title {\n color: #f87171;\n font-weight: 700;\n font-size: 0.95rem;\n display: flex;\n align-items: center;\n gap: 0.5rem;\n margin-bottom: 0.5rem;\n }\n\n #dijkstra-widget .alert-box-content {\n font-size: 0.85rem;\n line-height: 1.5;\n color: var(--text-main);\n }\n\n #dijkstra-widget .alert-box-content strong {\n color: #fca5a5;\n }\n<\/style>\n\n<div class=\"widget-container\" id=\"dijkstra-widget\" data-theme=\"dark\">\n <header>\n <h1>Dijkstra’s Algorithm Visualizer<\/h1>\n <p>Step-by-Step Graph Walkthrough for AlgoVelog<\/p>\n <\/header>\n\n <!-- Controls Panel -->\n <div class=\"card controls-card\">\n <div class=\"controls-wrapper\">\n <div class=\"btn-group\">\n <button id=\"btn-prev\" onclick=\"prevStep()\">\n <svg width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M15 18l-6-6 6-6\"><\/path><\/svg>\n Prev\n <\/button>\n <button id=\"btn-play\" class=\"primary\" onclick=\"togglePlay()\">\n <svg id=\"play-icon\" width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"currentColor\"><polygon points=\"5,3 19,12 5,21\"><\/polygon><\/svg>\n <span id=\"play-text\">Play<\/span>\n <\/button>\n <button id=\"btn-next\" onclick=\"nextStep()\">\n Next\n <svg width=\"16\" height=\"16\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M9 18l6-6-6-6\"><\/path><\/svg>\n <\/button>\n <button id=\"btn-reset\" onclick=\"resetVisualizer()\">\n Reset\n <\/button>\n <\/div>\n\n <div class=\"slider-container\">\n <span class=\"slider-label\" id=\"step-label\">Step 1 \/ 7<\/span>\n <input type=\"range\" id=\"step-slider\" min=\"0\" max=\"6\" value=\"0\" oninput=\"goToStep(parseInt(this.value))\">\n <\/div>\n <\/div>\n <\/div>\n\n <div class=\"visualizer-grid\">\n <!-- Graph Canvas Card -->\n <div class=\"card\">\n <div class=\"card-title\">\n <svg width=\"20\" height=\"20\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><circle cx=\"18\" cy=\"5\" r=\"3\"><\/circle><circle cx=\"6\" cy=\"12\" r=\"3\"><\/circle><circle cx=\"18\" cy=\"19\" r=\"3\"><\/circle><path d=\"M9 12h6M9 12l6-5M9 12l6 7\"><\/path><\/svg>\n Graph Layout\n <\/div>\n <div class=\"graph-container\">\n <svg viewBox=\"0 0 600 400\" id=\"graph-svg\">\n <!-- Edges (Rendered behind nodes) -->\n <g id=\"edges-group\">\n <!-- A-B (4) -->\n <line id=\"edge-A-B\" class=\"edge\" x1=\"100\" y1=\"100\" x2=\"300\" y2=\"100\"><\/line>\n <!-- A-C (2) -->\n <line id=\"edge-A-C\" class=\"edge\" x1=\"100\" y1=\"100\" x2=\"100\" y2=\"300\"><\/line>\n <!-- B-C (1) -->\n <line id=\"edge-B-C\" class=\"edge\" x1=\"300\" y1=\"100\" x2=\"100\" y2=\"300\"><\/line>\n <!-- B-D (5) -->\n <line id=\"edge-B-D\" class=\"edge\" x1=\"300\" y1=\"100\" x2=\"500\" y2=\"100\"><\/line>\n <!-- B-E (6) -->\n <line id=\"edge-B-E\" class=\"edge\" x1=\"300\" y1=\"100\" x2=\"300\" y2=\"300\"><\/line>\n <!-- C-D (8) -->\n <line id=\"edge-C-D\" class=\"edge\" x1=\"100\" y1=\"300\" x2=\"500\" y2=\"100\"><\/line>\n <!-- C-E (10) -->\n <line id=\"edge-C-E\" class=\"edge\" x1=\"100\" y1=\"300\" x2=\"300\" y2=\"300\"><\/line>\n <!-- D-E (2) -->\n <line id=\"edge-D-E\" class=\"edge\" x1=\"500\" y1=\"100\" x2=\"300\" y2=\"300\"><\/line>\n <\/g>\n\n <!-- Edge Weight Labels -->\n <g id=\"labels-group\">\n <!-- A-B (4) -->\n <g transform=\"translate(200, 100)\">\n <rect class=\"edge-label-bg\" x=\"-12\" y=\"-12\" width=\"24\" height=\"24\"><\/rect>\n <text id=\"weight-A-B\" class=\"edge-label\">4<\/text>\n <\/g>\n <!-- A-C (2) -->\n <g transform=\"translate(100, 200)\">\n <rect class=\"edge-label-bg\" x=\"-12\" y=\"-12\" width=\"24\" height=\"24\"><\/rect>\n <text id=\"weight-A-C\" class=\"edge-label\">2<\/text>\n <\/g>\n <!-- B-C (1) -->\n <g transform=\"translate(200, 200)\">\n <rect class=\"edge-label-bg\" x=\"-12\" y=\"-12\" width=\"24\" height=\"24\"><\/rect>\n <text id=\"weight-B-C\" class=\"edge-label\">1<\/text>\n <\/g>\n <!-- B-D (5) -->\n <g transform=\"translate(400, 100)\">\n <rect class=\"edge-label-bg\" x=\"-12\" y=\"-12\" width=\"24\" height=\"24\"><\/rect>\n <text id=\"weight-B-D\" class=\"edge-label\">5<\/text>\n <\/g>\n <!-- B-E (6) -->\n <g transform=\"translate(300, 200)\">\n <rect class=\"edge-label-bg\" x=\"-12\" y=\"-12\" width=\"24\" height=\"24\"><\/rect>\n <text id=\"weight-B-E\" class=\"edge-label\">6<\/text>\n <\/g>\n <!-- C-D (8) -->\n <g transform=\"translate(300, 200) translate(0, -35)\">\n <rect class=\"edge-label-bg\" x=\"-12\" y=\"-12\" width=\"24\" height=\"24\"><\/rect>\n <text id=\"weight-C-D\" class=\"edge-label\">8<\/text>\n <\/g>\n <!-- C-E (10) -->\n <g transform=\"translate(200, 300)\">\n <rect class=\"edge-label-bg\" x=\"-12\" y=\"-12\" width=\"24\" height=\"24\"><\/rect>\n <text id=\"weight-C-E\" class=\"edge-label\">10<\/text>\n <\/g>\n <!-- D-E (2) -->\n <g transform=\"translate(400, 200)\">\n <rect class=\"edge-label-bg\" x=\"-12\" y=\"-12\" width=\"24\" height=\"24\"><\/rect>\n <text id=\"weight-D-E\" class=\"edge-label\">2<\/text>\n <\/g>\n <\/g>\n\n <!-- Nodes (Rendered on top) -->\n <g id=\"nodes-group\">\n <!-- Node A -->\n <g id=\"node-A\" class=\"node\">\n <circle class=\"node-circle\" cx=\"100\" cy=\"100\" r=\"24\"><\/circle>\n <text class=\"node-text\" x=\"100\" y=\"100\">A<\/text>\n <\/g>\n <!-- Node B -->\n <g id=\"node-B\" class=\"node\">\n <circle class=\"node-circle\" cx=\"300\" cy=\"100\" r=\"24\"><\/circle>\n <text class=\"node-text\" x=\"300\" y=\"100\">B<\/text>\n <\/g>\n <!-- Node C -->\n <g id=\"node-C\" class=\"node\">\n <circle class=\"node-circle\" cx=\"100\" cy=\"300\" r=\"24\"><\/circle>\n <text class=\"node-text\" x=\"100\" y=\"300\">C<\/text>\n <\/g>\n <!-- Node D -->\n <g id=\"node-D\" class=\"node\">\n <circle class=\"node-circle\" cx=\"500\" cy=\"100\" r=\"24\"><\/circle>\n <text class=\"node-text\" x=\"500\" y=\"100\">D<\/text>\n <\/g>\n <!-- Node E -->\n <g id=\"node-E\" class=\"node\">\n <circle class=\"node-circle\" cx=\"300\" cy=\"300\" r=\"24\"><\/circle>\n <text class=\"node-text\" x=\"300\" y=\"300\">E<\/text>\n <\/g>\n <\/g>\n <\/svg>\n <\/div>\n <\/div>\n\n <!-- Algorithm State Card -->\n <div class=\"card\">\n <div class=\"card-title\">\n <svg width=\"20\" height=\"20\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M14 2H6a2 2 0 0 0-2 2v16a2 2 0 0 0 2 2h12a2 2 0 0 0 2-2V8z\"><\/path><polyline points=\"14 2 14 8 20 8\"><\/polyline><line x1=\"16\" y1=\"13\" x2=\"8\" y2=\"13\"><\/line><line x1=\"16\" y1=\"17\" x2=\"8\" y2=\"17\"><\/line><polyline points=\"10 9 9 9 8 9\"><\/polyline><\/svg>\n Step Execution & State\n <\/div>\n\n <h3 id=\"step-title\" style=\"margin-bottom: 0.5rem; font-family: 'Outfit';\">Step 1: Initialization<\/h3>\n <div id=\"step-description\" class=\"step-desc\">\n Set the distance to the source node A to 0, and all other nodes to infinity (\u221e). Add the source node to the Priority Queue.\n <\/div>\n\n <!-- Visited Status -->\n <div class=\"state-section\">\n <div class=\"state-section-title\">Visited Nodes<\/div>\n <div class=\"badge-list\" id=\"visited-container\">\n <span class=\"text-muted\" style=\"font-size: 0.9rem;\">None yet<\/span>\n <\/div>\n <\/div>\n\n <!-- Priority Queue Status -->\n <div class=\"state-section\">\n <div class=\"state-section-title\">Priority Queue (pq)<\/div>\n <div class=\"badge-list\" id=\"pq-container\">\n <!-- Badges dynamically added -->\n <\/div>\n <\/div>\n\n <!-- Distance and Parent Table -->\n <div class=\"state-section\">\n <div class=\"state-section-title\">Node Distances & Parents<\/div>\n <div style=\"background: rgba(15, 23, 42, 0.3); border-radius: 8px; border: 1px solid rgba(255, 255, 255, 0.04); overflow: hidden;\">\n <table>\n <thead>\n <tr>\n <th>Node<\/th>\n <th>Distance (dist)<\/th>\n <th>Parent<\/th>\n <\/tr>\n <\/thead>\n <tbody id=\"state-table-body\">\n <!-- Dynamically populated -->\n <\/tbody>\n <\/table>\n <\/div>\n <\/div>\n \n <!-- Path reconstruction (Only on final step) -->\n <div class=\"state-section\" id=\"final-path-section\" style=\"display: none;\">\n <div class=\"state-section-title\">Path Reconstruction to E<\/div>\n <div class=\"final-path-box\">\n <div><strong>Reconstruction:<\/strong> E ← B ← C ← A<\/div>\n <div><strong>Shortest Path:<\/strong> A → C → B → E (Cost: 9)<\/div>\n <\/div>\n <\/div>\n <\/div>\n <\/div>\n\n <!-- Advantages, Limitations and Critical Requirements Section -->\n <div class=\"info-grid\">\n <!-- Advantages Card -->\n <div class=\"card info-card-success\">\n <div class=\"card-title\">\n <svg width=\"20\" height=\"20\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"var(--color-success)\" stroke-width=\"2\"><path d=\"M22 11.08V12a10 10 0 1 1-5.93-9.14\"><\/path><polyline points=\"22 4 12 14.01 9 11.01\"><\/polyline><\/svg>\n Advantages & Strengths\n <\/div>\n <ul class=\"info-list\">\n <li>\n <span class=\"icon\">\u2713<\/span>\n <div><strong>Optimal solution guaranteed:<\/strong> Guaranteed for non-negative weights.<\/div>\n <\/li>\n <li>\n <span class=\"icon\">\u2713<\/span>\n <div><strong>Single-source problem:<\/strong> Finds shortest paths to all vertices in one run.<\/div>\n <\/li>\n <li>\n <span class=\"icon\">\u2713<\/span>\n <div><strong>Greedy approach:<\/strong> Greedy approach with proven correctness.<\/div>\n <\/li>\n <li>\n <span class=\"icon\">\u2713<\/span>\n <div><strong>Efficient time complexity:<\/strong> Efficient O((V+E) log V) with binary heap.<\/div>\n <\/li>\n <li>\n <span class=\"icon\">\u2713<\/span>\n <div><strong>Practical use:<\/strong> Practical and widely implemented in real systems.<\/div>\n <\/li>\n <\/ul>\n <\/div>\n\n <!-- Limitations Card -->\n <div class=\"card info-card-warning\">\n <div class=\"card-title\">\n <svg width=\"20\" height=\"20\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"var(--color-danger)\" stroke-width=\"2\"><circle cx=\"12\" cy=\"12\" r=\"10\"><\/circle><line x1=\"15\" y1=\"9\" x2=\"9\" y2=\"15\"><\/line><line x1=\"9\" y1=\"9\" x2=\"15\" y2=\"15\"><\/line><\/svg>\n Limitations & Constraints\n <\/div>\n <ul class=\"info-list\">\n <li>\n <span class=\"icon\">\u2717<\/span>\n <div><strong>Non-negative weights only:<\/strong> Fails with negative edge weights.<\/div>\n <\/li>\n <li>\n <span class=\"icon\">\u2717<\/span>\n <div><strong>All-pairs limitation:<\/strong> Not suitable for finding all-pairs shortest paths – use Floyd-Warshall for that.<\/div>\n <\/li>\n <li>\n <span class=\"icon\">\u2717<\/span>\n <div><strong>Source-biased:<\/strong> Must run separately from each source.<\/div>\n <\/li>\n <li>\n <span class=\"icon\">\u2717<\/span>\n <div><strong>Dense graphs:<\/strong> O(V\u00b2) with linear search may be faster for very dense graphs.<\/div>\n <\/li>\n <\/ul>\n\n <!-- Critical Requirement Alert -->\n <div class=\"alert-box\">\n <div class=\"alert-box-title\" style=\"color: #ef4444;\">\n <svg width=\"18\" height=\"18\" viewBox=\"0 0 24 24\" fill=\"none\" stroke=\"currentColor\" stroke-width=\"2\"><path d=\"M10.29 3.86L1.82 18a2 2 0 0 0 1.71 3h16.94a2 2 0 0 0 1.71-3L13.71 3.86a2 2 0 0 0-3.42 0z\"><\/path><line x1=\"12\" y1=\"9\" x2=\"12\" y2=\"13\"><\/line><line x1=\"12\" y1=\"17\" x2=\"12.01\" y2=\"17\"><\/line><\/svg>\n Critical Requirement\n <\/div>\n <div class=\"alert-box-content\">\n <p style=\"margin-bottom: 0.5rem; font-weight: 600;\">All edge weights MUST be non-negative. If even one negative edge exists, Dijkstra produces incorrect results.<\/p>\n <p style=\"margin-bottom: 0.5rem; font-size: 0.8rem; color: var(--text-muted);\">\n <strong>Why?<\/strong> Dijkstra’s proof of correctness relies on: once a vertex is finalized (marked as visited), its distance is optimal. With negative weights, a later-discovered path through a negative edge could make a “finalized” distance suboptimal. Dijkstra cannot backtrack to reconsider finalized vertices.\n <\/p>\n <p style=\"font-size: 0.8rem;\">\n <strong>Alternative for negative weights:<\/strong> Use <strong>Bellman-Ford algorithm<\/strong> (O(VE) time, can detect negative cycles).\n <\/p>\n <\/div>\n <\/div>\n <\/div>\n <\/div>\n\n<\/div>\n\n<script>\n \/\/ Data defining each step of the Dijkstra run\n const steps = [\n {\n title: \"Step 1: Initialization\",\n desc: \"Initialize distance to source node A as 0, and all other nodes to infinity (∞). Add the pair (0, A) to the priority queue.\",\n visited: [],\n processing: null,\n targets: [\"A\"],\n dist: {A: \"0\", B: \"∞\", C: \"∞\", D: \"∞\", E: \"∞\"},\n parent: {A: \"null\", B: \"null\", C: \"null\", D: \"null\", E: \"null\"},\n pq: [\"(0, A)\"],\n activeEdges: [],\n updatingEdges: [],\n shortestPathEdges: []\n },\n {\n title: \"Step 2: Process A\",\n desc: \"Extract (0, A) from the priority queue. Mark A as visited.<br>Explore A's neighbors:<br>\u2022 <strong>B:<\/strong> 0 + 4 = 4 < ∞. Update dist[B] = 4, parent[B] = A.<br>\u2022 <strong>C:<\/strong> 0 + 2 = 2 < ∞. Update dist[C] = 2, parent[C] = A.\",\n visited: [\"A\"],\n processing: \"A\",\n targets: [\"B\", \"C\"],\n dist: {A: \"0\", B: \"4\", C: \"2\", D: \"∞\", E: \"∞\"},\n parent: {A: \"null\", B: \"A\", C: \"A\", D: \"null\", E: \"null\"},\n pq: [\"(2, C)\", \"(4, B)\"],\n activeEdges: [\"edge-A-B\", \"edge-A-C\"],\n updatingEdges: [\"edge-A-B\", \"edge-A-C\"],\n shortestPathEdges: []\n },\n {\n title: \"Step 3: Process C\",\n desc: \"Extract (2, C) from the priority queue. Mark C as visited.<br>Explore C's neighbors:<br>\u2022 <strong>A:<\/strong> Already visited, skip.<br>\u2022 <strong>B:<\/strong> 2 + 1 = 3 < 4. Update dist[B] = 3, parent[B] = C.<br>\u2022 <strong>D:<\/strong> 2 + 8 = 10 < ∞. Update dist[D] = 10, parent[D] = C.<br>\u2022 <strong>E:<\/strong> 2 + 10 = 12 < ∞. Update dist[E] = 12, parent[E] = C.\",\n visited: [\"A\", \"C\"],\n processing: \"C\",\n targets: [\"B\", \"D\", \"E\"],\n dist: {A: \"0\", B: \"3\", C: \"2\", D: \"10\", E: \"12\"},\n parent: {A: \"null\", B: \"C\", C: \"A\", D: \"C\", E: \"C\"},\n pq: [\"(3, B)\", \"(4, B)\", \"(10, D)\", \"(12, E)\"],\n activeEdges: [\"edge-A-C\", \"edge-B-C\", \"edge-C-D\", \"edge-C-E\"],\n updatingEdges: [\"edge-B-C\", \"edge-C-D\", \"edge-C-E\"],\n shortestPathEdges: []\n },\n {\n title: \"Step 4: Process B\",\n desc: \"Extract (3, B) from the priority queue. Mark B as visited.<br>Explore B's neighbors:<br>\u2022 <strong>A, C:<\/strong> Already visited, skip.<br>\u2022 <strong>D:<\/strong> 3 + 5 = 8 < 10. Update dist[D] = 8, parent[D] = B.<br>\u2022 <strong>E:<\/strong> 3 + 6 = 9 < 12. Update dist[E] = 9, parent[E] = B.\",\n visited: [\"A\", \"C\", \"B\"],\n processing: \"B\",\n targets: [\"D\", \"E\"],\n dist: {A: \"0\", B: \"3\", C: \"2\", D: \"8\", E: \"9\"},\n parent: {A: \"null\", B: \"C\", C: \"A\", D: \"B\", E: \"B\"},\n pq: [\"(4, B)\", \"(8, D)\", \"(9, E)\", \"(10, D)\", \"(12, E)\"],\n activeEdges: [\"edge-A-C\", \"edge-B-C\", \"edge-B-D\", \"edge-B-E\"],\n updatingEdges: [\"edge-B-D\", \"edge-B-E\"],\n shortestPathEdges: []\n },\n {\n title: \"Step 5: Process D\",\n desc: \"Extract (4, B) from the priority queue. Since B is already visited, skip it.<br>Extract (8, D) from the priority queue. Mark D as visited.<br>Explore D's neighbors:<br>\u2022 <strong>B, C:<\/strong> Already visited, skip.<br>\u2022 <strong>E:<\/strong> 8 + 2 = 10 > 9. No update is made since 9 is already shorter.\",\n visited: [\"A\", \"C\", \"B\", \"D\"],\n processing: \"D\",\n targets: [\"E\"],\n dist: {A: \"0\", B: \"3\", C: \"2\", D: \"8\", E: \"9\"},\n parent: {A: \"null\", B: \"C\", C: \"A\", D: \"B\", E: \"B\"},\n pq: [\"(9, E)\", \"(10, D)\", \"(12, E)\"],\n activeEdges: [\"edge-A-C\", \"edge-B-C\", \"edge-B-D\", \"edge-B-E\", \"edge-D-E\"],\n updatingEdges: [],\n shortestPathEdges: []\n },\n {\n title: \"Step 6: Process E\",\n desc: \"Extract (9, E) from the priority queue. Mark E as visited.<br>E has no unvisited neighbors. The remaining elements in the priority queue will be skipped when extracted as their destinations are already visited.\",\n visited: [\"A\", \"C\", \"B\", \"D\", \"E\"],\n processing: \"E\",\n targets: [],\n dist: {A: \"0\", B: \"3\", C: \"2\", D: \"8\", E: \"9\"},\n parent: {A: \"null\", B: \"C\", C: \"A\", D: \"B\", E: \"B\"},\n pq: [],\n activeEdges: [\"edge-A-C\", \"edge-B-C\", \"edge-B-D\", \"edge-B-E\"],\n updatingEdges: [],\n shortestPathEdges: []\n },\n {\n title: \"Step 7: Final Result\",\n desc: \"The algorithm has completed! The shortest paths are fully reconstructed. The shortest path from A to E is highlighted in blue: A → C → B → E (Total cost = 9).\",\n visited: [\"A\", \"C\", \"B\", \"D\", \"E\"],\n processing: null,\n targets: [],\n dist: {A: \"0\", B: \"3\", C: \"2\", D: \"8\", E: \"9\"},\n parent: {A: \"null\", B: \"C\", C: \"A\", D: \"B\", E: \"B\"},\n pq: [],\n activeEdges: [],\n updatingEdges: [],\n shortestPathEdges: [\"edge-A-C\", \"edge-B-C\", \"edge-B-E\"]\n }\n ];\n\n let currentStep = 0;\n let playInterval = null;\n const playSpeed = 2500; \/\/ ms per step\n\n \/\/ Initialize display\n goToStep(0);\n\n function goToStep(stepIndex) {\n if (stepIndex < 0 || stepIndex >= steps.length) return;\n \n \/\/ Check if values updated from previous step to trigger animation\n const oldStep = currentStep;\n currentStep = stepIndex;\n \n document.getElementById(\"step-slider\").value = currentStep;\n document.getElementById(\"step-label\").innerText = `Step ${currentStep + 1} \/ ${steps.length}`;\n \n \/\/ Update button states\n document.getElementById(\"btn-prev\").disabled = (currentStep === 0);\n document.getElementById(\"btn-next\").disabled = (currentStep === steps.length - 1);\n \n const stepData = steps[currentStep];\n const prevStepData = oldStep !== currentStep ? steps[oldStep] : null;\n\n \/\/ Update Text Content\n document.getElementById(\"step-title\").innerText = stepData.title;\n document.getElementById(\"step-description\").innerHTML = stepData.desc;\n \n const descDiv = document.getElementById(\"step-description\");\n if (stepData.processing) {\n descDiv.className = \"step-desc\";\n } else if (stepData.visited.length === 5) {\n descDiv.className = \"step-desc visited-update\";\n } else {\n descDiv.className = \"step-desc\";\n }\n\n \/\/ Update Visited Container\n const visitedContainer = document.getElementById(\"visited-container\");\n if (stepData.visited.length === 0) {\n visitedContainer.innerHTML = `<span class=\"text-muted\" style=\"font-size: 0.9rem;\">None yet<\/span>`;\n } else {\n visitedContainer.innerHTML = stepData.visited.map(node => \n `<span class=\"badge visited\">${node}<\/span>`\n ).join(\"\");\n }\n\n \/\/ Update Priority Queue Container\n const pqContainer = document.getElementById(\"pq-container\");\n if (stepData.pq.length === 0) {\n pqContainer.innerHTML = `<span class=\"text-muted\" style=\"font-size: 0.9rem;\">Empty<\/span>`;\n } else {\n pqContainer.innerHTML = stepData.pq.map(item => \n `<span class=\"badge queue-item\">${item}<\/span>`\n ).join(\"\");\n }\n\n \/\/ Populate Table\n const tableBody = document.getElementById(\"state-table-body\");\n tableBody.innerHTML = \"\";\n [\"A\", \"B\", \"C\", \"D\", \"E\"].forEach(node => {\n const distVal = stepData.dist[node];\n const parentVal = stepData.parent[node];\n \n \/\/ Determine if this cell's values changed compared to previous step to highlight it\n let distClass = \"val-cell\";\n let parentClass = \"val-cell\";\n if (prevStepData && prevStepData.dist[node] !== distVal) {\n distClass += \" updated-val\";\n }\n if (prevStepData && prevStepData.parent[node] !== parentVal) {\n parentClass += \" updated-val\";\n }\n\n const row = document.createElement(\"tr\");\n row.innerHTML = `\n <td style=\"font-weight: 600;\">Node ${node}<\/td>\n <td class=\"${distClass}\">${distVal}<\/td>\n <td class=\"${parentClass}\">${parentVal}<\/td>\n `;\n tableBody.appendChild(row);\n });\n\n \/\/ Toggle Final Path Section\n const finalPathSection = document.getElementById(\"final-path-section\");\n if (currentStep === steps.length - 1) {\n finalPathSection.style.display = \"block\";\n } else {\n finalPathSection.style.display = \"none\";\n }\n\n \/\/ Update SVG Elements class names\n \/\/ 1. Nodes\n [\"A\", \"B\", \"C\", \"D\", \"E\"].forEach(nodeId => {\n const nodeEl = document.getElementById(`node-${nodeId}`);\n if (nodeEl) {\n nodeEl.className.baseVal = \"node\"; \/\/ Reset\n \n if (stepData.processing === nodeId) {\n nodeEl.className.baseVal = \"node processing\";\n } else if (stepData.visited.includes(nodeId)) {\n nodeEl.className.baseVal = \"node visited\";\n } else if (stepData.targets.includes(nodeId)) {\n nodeEl.className.baseVal = \"node target\";\n }\n }\n });\n\n \/\/ 2. Edges\n const allEdges = [\"edge-A-B\", \"edge-A-C\", \"edge-B-C\", \"edge-B-D\", \"edge-B-E\", \"edge-C-D\", \"edge-C-E\", \"edge-D-E\"];\n allEdges.forEach(edgeId => {\n const edgeEl = document.getElementById(edgeId);\n if (edgeEl) {\n const labelId = edgeId.replace(\"edge\", \"weight\");\n const labelEl = document.getElementById(labelId);\n \n edgeEl.className.baseVal = \"edge\"; \/\/ Reset\n if (labelEl) labelEl.className.baseVal = \"edge-label\"; \/\/ Reset\n\n if (stepData.shortestPathEdges.includes(edgeId)) {\n edgeEl.className.baseVal = \"edge shortest-path\";\n if (labelEl) labelEl.className.baseVal = \"edge-label active\";\n } else if (stepData.updatingEdges.includes(edgeId)) {\n edgeEl.className.baseVal = \"edge updating\";\n if (labelEl) labelEl.className.baseVal = \"edge-label active\";\n } else if (stepData.activeEdges.includes(edgeId)) {\n edgeEl.className.baseVal = \"edge active\";\n if (labelEl) labelEl.className.baseVal = \"edge-label active\";\n }\n }\n });\n }\n\n function nextStep() {\n if (currentStep < steps.length - 1) {\n goToStep(currentStep + 1);\n } else {\n pause();\n }\n }\n\n function prevStep() {\n if (currentStep > 0) {\n goToStep(currentStep - 1);\n }\n }\n\n function resetVisualizer() {\n pause();\n goToStep(0);\n }\n\n function togglePlay() {\n if (playInterval) {\n pause();\n } else {\n play();\n }\n }\n\n function play() {\n if (currentStep === steps.length - 1) {\n goToStep(0); \/\/ Loop back to start if at end\n }\n \n document.getElementById(\"play-text\").innerText = \"Pause\";\n \/\/ Change SVG to pause bars\n document.getElementById(\"play-icon\").innerHTML = `<rect x=\"5\" y=\"4\" width=\"4\" height=\"16\"\/><rect x=\"15\" y=\"4\" width=\"4\" height=\"16\"\/>`;\n \n playInterval = setInterval(nextStep, playSpeed);\n }\n\n function pause() {\n if (playInterval) {\n clearInterval(playInterval);\n playInterval = null;\n }\n document.getElementById(\"play-text\").innerText = \"Play\";\n \/\/ Change SVG back to play triangle\n document.getElementById(\"play-icon\").innerHTML = `<polygon points=\"5,3 19,12 5,21\"\/>`;\n }\n<\/script>\n}\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Dijkstra’s Algorithm: A Comprehensive Guide Before directly jumping to Dijkstra’s Algorithm, let’s first understand some basic concepts related to it. Weighted Graph A weighted graph is a network of vertices (nodes) connected by edges, where each edge has a weight. The weight is a numerical value representing cost, distance, time, capacity, or any other metric […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-27","post","type-post","status-publish","format-standard","hentry","category-dsa"],"_links":{"self":[{"href":"https:\/\/algovelgo.com\/index.php?rest_route=\/wp\/v2\/posts\/27","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/algovelgo.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/algovelgo.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/algovelgo.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/algovelgo.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=27"}],"version-history":[{"count":0,"href":"https:\/\/algovelgo.com\/index.php?rest_route=\/wp\/v2\/posts\/27\/revisions"}],"wp:attachment":[{"href":"https:\/\/algovelgo.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=27"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/algovelgo.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=27"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/algovelgo.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=27"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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