From b1e07e8aa3218f8dd90d48fe6144de52b45628b2 Mon Sep 17 00:00:00 2001
From: rhombus2002 <rkocharyan2002@gmail.com>
Date: Tue, 24 Jun 2025 16:45:43 +0400
Subject: [PATCH 1/3] Update euler_path.md

---
 src/graph/euler_path.md | 138 +++++++++++++++++++++-------------------
 1 file changed, 71 insertions(+), 67 deletions(-)

diff --git a/src/graph/euler_path.md b/src/graph/euler_path.md
index a54906253..7da17d983 100644
--- a/src/graph/euler_path.md
+++ b/src/graph/euler_path.md
@@ -66,97 +66,101 @@ The program below searches for and outputs a Eulerian loop or path in a graph, o
 
 First, the program checks the degree of vertices: if there are no vertices with an odd degree, then the graph has an Euler cycle, if there are $2$ vertices with an odd degree, then in the graph there is only an Euler path (but no Euler cycle), if there are more than $2$ such vertices, then in the graph there is no Euler cycle or Euler path.
 To find the Euler path (not a cycle), let's do this: if $V1$ and $V2$ are two vertices of odd degree, then just add an edge $(V1, V2)$, in the resulting graph we find the Euler cycle (it will obviously exist), and then remove the "fictitious" edge $(V1, V2)$ from the answer.
-We will look for the Euler cycle exactly as described above (non-recursive version), and at the same time at the end of this algorithm we will check whether the graph was connected or not (if the graph was not connected, then at the end of the algorithm some edges will remain in the graph, and in this case we need to print $-1$).
+We will look for the Euler cycle exactly as described above (recursive version), and at the same time at the end of this algorithm we will check whether the graph was connected or not (if the graph was not connected, then at the end of the algorithm some edges will remain in the graph, and in this case we need to print $-1$).
 Finally, the program takes into account that there can be isolated vertices in the graph.
 
-Notice that we use an adjacency matrix in this problem.
-Also this implementation handles finding the next with brute-force, which requires to iterate over the complete row in the matrix over and over.
-A better way would be to store the graph as an adjacency list, and remove edges in $O(1)$ and mark the reversed edges in separate list.
-This way we can achieve an $O(N)$ algorithm.
-
 ```cpp
-int main() {
-    int n;
-    vector<vector<int>> g(n, vector<int>(n));
-    // reading the graph in the adjacency matrix
-
-    vector<int> deg(n);
-    for (int i = 0; i < n; ++i) {
-        for (int j = 0; j < n; ++j)
-            deg[i] += g[i][j];
-    }
+vector<pair<int,int>> edges;
+vector<vector<int>> g;
+vector<bool> used;
+vector<int> res;
+
+void add_edge(int u, int v) {
+    int idx = (int) edges.size();
+    edges.emplace_back(u, v);
+    g[u].push_back(idx);
+    g[v].push_back(idx);
+}
 
-    int first = 0;
-    while (first < n && !deg[first])
-        ++first;
-    if (first == n) {
-        cout << -1;
-        return 0;
+void dfs(int v) {
+    while (!g[v].empty()) {
+        int idx = g[v].back();
+        g[v].pop_back();
+        if (used[idx]) continue;
+        used[idx] = true;
+        auto [u, w] = edges[idx];
+        dfs(u ^ w ^ v);
     }
+    res.push_back(v);
+}
 
+int main() {
+    ios::sync_with_stdio(0);
+    cin.tie(0);
+    int n, m;
+    cin >> n >> m;
+    edges.clear();
+    g.assign(n, vector<int> ());
+    used.assign(m + 1, false);
+    res.clear();
+    for (int i = 0; i < m; i++) {
+        int u, v;
+        cin >> u >> v;
+        u--, v--;
+        add_edge(u, v);
+    }
     int v1 = -1, v2 = -1;
     bool bad = false;
-    for (int i = 0; i < n; ++i) {
-        if (deg[i] & 1) {
-            if (v1 == -1)
+    for (int i = 0; i < n; i++) {
+        if ((int) g[i].size() % 2) {
+            if (v1 == -1) {
                 v1 = i;
-            else if (v2 == -1)
+            } else if (v2 == -1) {
                 v2 = i;
-            else
+            } else {
                 bad = true;
+            }
         }
     }
-
-    if (v1 != -1)
-        ++g[v1][v2], ++g[v2][v1];
-
-    stack<int> st;
-    st.push(first);
-    vector<int> res;
-    while (!st.empty()) {
-        int v = st.top();
-        int i;
-        for (i = 0; i < n; ++i)
-            if (g[v][i])
-                break;
-        if (i == n) {
-            res.push_back(v);
-            st.pop();
-        } else {
-            --g[v][i];
-            --g[i][v];
-            st.push(i);
-        }
+    int first = 0;
+    while (first < n && g[first].empty()) {
+        first++;
+    }
+    if (bad || (v1 != -1 && v2 == -1) || first == n) {
+        cout << -1 << '\n';
+        return 0;
     }
-
     if (v1 != -1) {
-        for (size_t i = 0; i + 1 < res.size(); ++i) {
+        add_edge(v1, v2);
+    }
+    dfs(first);
+    if (v1 != -1) {
+        for (int i = 0; i + 1 < (int) res.size(); i++) {
             if ((res[i] == v1 && res[i + 1] == v2) ||
                 (res[i] == v2 && res[i + 1] == v1)) {
-                vector<int> res2;
-                for (size_t j = i + 1; j < res.size(); ++j)
-                    res2.push_back(res[j]);
-                for (size_t j = 1; j <= i; ++j)
-                    res2.push_back(res[j]);
-                res = res2;
+                vector<int> nw;
+                for (int j = i + 1; j < (int) res.size(); j++) {
+                    nw.push_back(res[j]);
+                }
+                for (int j = 1; j <= i; j++) {
+                    nw.push_back(res[j]);
+                }
+                res = nw;
                 break;
             }
         }
     }
-
-    for (int i = 0; i < n; ++i) {
-        for (int j = 0; j < n; ++j) {
-            if (g[i][j])
-                bad = true;
+    for (int i = 0; i < n; i++) {
+        if (!g[i].empty()) {
+            cout << -1 << '\n';
+            return 0;
         }
     }
-
-    if (bad) {
-        cout << -1;
-    } else {
-        for (int x : res)
-            cout << x << " ";
+    for (int x : res) {
+        cout << x + 1 << ' ';
     }
+    cout << '\n';
+    return 0;
 }
 ```
 ### Practice problems:

From aba953bfacaa3149dd06292fbdfc599171263e62 Mon Sep 17 00:00:00 2001
From: rhombus2002 <rkocharyan2002@gmail.com>
Date: Tue, 24 Jun 2025 18:32:49 +0400
Subject: [PATCH 2/3] Small changes in code

---
 src/graph/euler_path.md | 21 ++++++++++-----------
 1 file changed, 10 insertions(+), 11 deletions(-)

diff --git a/src/graph/euler_path.md b/src/graph/euler_path.md
index 7da17d983..e6e956979 100644
--- a/src/graph/euler_path.md
+++ b/src/graph/euler_path.md
@@ -98,17 +98,8 @@ int main() {
     ios::sync_with_stdio(0);
     cin.tie(0);
     int n, m;
-    cin >> n >> m;
-    edges.clear();
-    g.assign(n, vector<int> ());
-    used.assign(m + 1, false);
-    res.clear();
-    for (int i = 0; i < m; i++) {
-        int u, v;
-        cin >> u >> v;
-        u--, v--;
-        add_edge(u, v);
-    }
+    // reading the graph
+
     int v1 = -1, v2 = -1;
     bool bad = false;
     for (int i = 0; i < n; i++) {
@@ -122,18 +113,24 @@ int main() {
             }
         }
     }
+
     int first = 0;
     while (first < n && g[first].empty()) {
         first++;
     }
+
     if (bad || (v1 != -1 && v2 == -1) || first == n) {
         cout << -1 << '\n';
         return 0;
     }
+
     if (v1 != -1) {
         add_edge(v1, v2);
     }
+
+    used.assign((int) edges.size(), false);
     dfs(first);
+
     if (v1 != -1) {
         for (int i = 0; i + 1 < (int) res.size(); i++) {
             if ((res[i] == v1 && res[i + 1] == v2) ||
@@ -150,12 +147,14 @@ int main() {
             }
         }
     }
+
     for (int i = 0; i < n; i++) {
         if (!g[i].empty()) {
             cout << -1 << '\n';
             return 0;
         }
     }
+
     for (int x : res) {
         cout << x + 1 << ' ';
     }

From 674c5326f3d376edb73fdc738b9157448d8b1b80 Mon Sep 17 00:00:00 2001
From: rhombus2002 <rkocharyan2002@gmail.com>
Date: Fri, 27 Jun 2025 16:44:08 +0400
Subject: [PATCH 3/3] Non-recursive implementation

---
 src/graph/euler_path.md | 33 +++++++++++++++++++--------------
 1 file changed, 19 insertions(+), 14 deletions(-)

diff --git a/src/graph/euler_path.md b/src/graph/euler_path.md
index e6e956979..c22e1f00f 100644
--- a/src/graph/euler_path.md
+++ b/src/graph/euler_path.md
@@ -66,7 +66,7 @@ The program below searches for and outputs a Eulerian loop or path in a graph, o
 
 First, the program checks the degree of vertices: if there are no vertices with an odd degree, then the graph has an Euler cycle, if there are $2$ vertices with an odd degree, then in the graph there is only an Euler path (but no Euler cycle), if there are more than $2$ such vertices, then in the graph there is no Euler cycle or Euler path.
 To find the Euler path (not a cycle), let's do this: if $V1$ and $V2$ are two vertices of odd degree, then just add an edge $(V1, V2)$, in the resulting graph we find the Euler cycle (it will obviously exist), and then remove the "fictitious" edge $(V1, V2)$ from the answer.
-We will look for the Euler cycle exactly as described above (recursive version), and at the same time at the end of this algorithm we will check whether the graph was connected or not (if the graph was not connected, then at the end of the algorithm some edges will remain in the graph, and in this case we need to print $-1$).
+We will look for the Euler cycle exactly as described above (non-recursive version), and at the same time at the end of this algorithm we will check whether the graph was connected or not (if the graph was not connected, then at the end of the algorithm some edges will remain in the graph, and in this case we need to print $-1$).
 Finally, the program takes into account that there can be isolated vertices in the graph.
 
 ```cpp
@@ -82,18 +82,6 @@ void add_edge(int u, int v) {
     g[v].push_back(idx);
 }
 
-void dfs(int v) {
-    while (!g[v].empty()) {
-        int idx = g[v].back();
-        g[v].pop_back();
-        if (used[idx]) continue;
-        used[idx] = true;
-        auto [u, w] = edges[idx];
-        dfs(u ^ w ^ v);
-    }
-    res.push_back(v);
-}
-
 int main() {
     ios::sync_with_stdio(0);
     cin.tie(0);
@@ -129,7 +117,24 @@ int main() {
     }
 
     used.assign((int) edges.size(), false);
-    dfs(first);
+    
+    stack <int> s;
+    s.push(first);
+    while (!s.empty()) {
+        int v = s.top();
+        if (g[v].empty()) {
+            res.push_back(v);
+            s.pop();
+            continue;
+        }
+        int idx = g[v].back();
+        g[v].pop_back();
+        if (used[idx]) continue;
+        used[idx] = true;
+        auto [u, w] = edges[idx];
+        int nxt = v ^ u ^ w;
+        s.push(nxt);
+    }
 
     if (v1 != -1) {
         for (int i = 0; i + 1 < (int) res.size(); i++) {

<!DOCTYPE html PUBLIC '-//W3C//DTD XHTML 1.0 Transitional//EN' 'http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd'>
<html xmlns='http://www.w3.org/1999/xhtml'>
<head>
<title>pFad - Phonifier reborn</title>
<meta http-equiv='Content-Type' content='text/html; charset=utf-8' />
</head>
<body>
<h1>Pfad - The Proxy pFad of &#169; 2024 Garber Painting. All rights reserved.</h1>


<!-- Disclaimer -->
<p>Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.</p>
<br>
<p>Alternative Proxies:</p><p><a href="http://rainy.clevelandohioweatherforecast.com/php-proxy/index.php?q=https://patch-diff.githubusercontent.com/raw/cp-algorithms/cp-algorithms/pull/1472.patch" target="_blank">Alternative Proxy</a></p><p><a href="http://rainy.clevelandohioweatherforecast.com/pFad/index.php?u=https://patch-diff.githubusercontent.com/raw/cp-algorithms/cp-algorithms/pull/1472.patch" target="_blank">pFad Proxy</a></p><p><a href="http://rainy.clevelandohioweatherforecast.com/pFad/v3index.php?u=https://patch-diff.githubusercontent.com/raw/cp-algorithms/cp-algorithms/pull/1472.patch" target="_blank">pFad v3 Proxy</a></p><p><a href="http://rainy.clevelandohioweatherforecast.com/pFad/v4index.php?u=https://patch-diff.githubusercontent.com/raw/cp-algorithms/cp-algorithms/pull/1472.patch" target="_blank">pFad v4 Proxy</a></p></body>
</html>