cp-algorithms · clumsy · Sep 23, 2024 · Aug 26, 2025 · Aug 26, 2025 · Aug 26, 2025
diff --git a/src/graph/01_bfs.md b/src/graph/01_bfs.md
@@ -15,29 +15,44 @@ In this article we demonstrate how we can use BFS to solve the SSSP (single-sour
 ## Algorithm
 
 We can develop the algorithm by closely studying Dijkstra's algorithm and thinking about the consequences that our special graph implies.
-The general form of Dijkstra's algorithm is (here a `set` is used for the priority queue):
-
-```cpp
-d.assign(n, INF);
-d[s] = 0;
-set<pair<int, int>> q;
-q.insert({0, s});
-while (!q.empty()) {
-    int v = q.begin()->second;
-    q.erase(q.begin());
-
-    for (auto edge : adj[v]) {
-        int u = edge.first;
-        int w = edge.second;
-
-        if (d[v] + w < d[u]) {
-            q.erase({d[u], u});
-            d[u] = d[v] + w;
-            q.insert({d[u], u});
+The general form of Dijkstra's algorithm is:
+
+=== "C++"
+    ```cpp
+    int n = adj.size();
+    d.assign(n, INF);
+    d[s] = 0;
+    priority_queue<pair<int, int>,vector<pair<int,int>>, greater<pair<int,int>> > q;
+    q.push({0,s});
+    while (!q.empty()) {
+        auto [dv, v] = q.top();
+        q.pop();
+        if (dv == d[v]) {
+            for (auto [u, w] : adj[v]) {
+                if (d[v] + w < d[u]) {
+                    d[u] = d[v] + w;
+                    q.push({d[u], u});
+                }
+            }
         }
     }
-}
-```
+    ```
+=== "Python"
+    ```py
+    from heapq import heappop, heappush
+
+
+    d = [float("inf")] * n
+    d[s] = 0
+    q = [(0, s)]
+    while q:
+        dv, v = heappop(q)
+        if dv == d[v]:
+            for u, w in adj[v]:
+                if d[v] + w < d[u]:
+                    d[u] = d[v] + w
+                    heappush(q, (d[u], u))
+    ```
 
 We can notice that the difference between the distances between the source `s` and two other vertices in the queue differs by at most one.
 Especially, we know that $d[v] \le d[u] \le d[v] + 1$ for each $u \in Q$.
@@ -53,27 +68,40 @@ This structure is so simple, that we don't need an actual priority queue, i.e. u
 We can simply use a normal queue, and append new vertices at the beginning if the corresponding edge has weight $0$, i.e. if $d[u] = d[v]$, or at the end if the edge has weight $1$, i.e. if $d[u] = d[v] + 1$.
 This way the queue still remains sorted at all time.
 
-```cpp
-vector<int> d(n, INF);
-d[s] = 0;
-deque<int> q;
-q.push_front(s);
-while (!q.empty()) {
-    int v = q.front();
-    q.pop_front();
-    for (auto edge : adj[v]) {
-        int u = edge.first;
-        int w = edge.second;
-        if (d[v] + w < d[u]) {
-            d[u] = d[v] + w;
-            if (w == 1)
-                q.push_back(u);
-            else
-                q.push_front(u);
+=== "C++"
+    ```cpp
+    vector<int> d(n, INF);
+    d[s] = 0;
+    deque<int> q = {s};
+    while (!empty(q)) {
+        int v = q.front();
+        q.pop_front();
+        for (auto [u, w] : adj[v]) {
+            if (d[v] + w < d[u]) {
+                d[u] = d[v] + w;
+                if (w == 1)
+                    q.push_back(u);
+                else
+                    q.push_front(u);
+            }
         }
     }
-}
-```
+    ```
+=== "Python"
+    ```py
+    d = [float("inf")] * n
+    d[s] = 0
+    q = deque([s])
+    while q:
+        v = q.popleft()
+        for u, w in adj[v]:
+            if d[v] + w < d[u]:
+                d[u] = d[v] + w
+                if w == 1:
+                    q.append(u)
+                else:
+                    q.appendleft(u)
+    ```
 
 ## Dial's algorithm