modify graphs a bit, corrections
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Benjamin Qi
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4 changed files with 58 additions and 40 deletions
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# Silver - Graphs
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# Silver - DFS
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Author: Siyong Huang
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@ -12,21 +12,24 @@ Author: Siyong Huang
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## Prerequisites
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- [Graph Theory](https://csacademy.com/lesson/introduction_to_graphs)
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- [Graph Representations](https://csacademy.com/lesson/graph_representation)
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- [CSAcademy Graph Theory](https://csacademy.com/lesson/introduction_to_graphs)
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- [CSAcademy Graph Representations](https://csacademy.com/lesson/graph_representation)
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- Note: DFS is most commonly implemented with adjacency lists
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## Depth First Search (DFS)
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*Depth First Search*, more commonly DFS, is a fundamental graph algorithm that traverses an entire connected component. The rest of this document describe various applications of DFS.
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*Depth First Search*, more commonly DFS, is a fundamental graph algorithm that traverses an entire connected component. The rest of this document describes various applications of DFS. Of course, it is one possible way to implement flood fill.
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- [CSES Building Roads](https://cses.fi/problemset/task/1666)
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### Tutorial
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- Recommended:
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- [CSAcademy BFS](https://csacademy.com/lesson/depth_first_search/)
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- [CSAcademy DFS](https://csacademy.com/lesson/depth_first_search/)
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- Additional:
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- CPH Chapter 12
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- [cp-algo DFS](https://cp-algorithms.com/graph/depth-first-search.html)
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- hard to parse if this is your first time learning about DFS
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### Problems
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@ -38,7 +41,10 @@ Author: Siyong Huang
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## Flood Fill
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*Flood Fill* refers to finding the number of connected components in a graph, frequently on a grid.
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*Flood Fill* refers to finding the number of connected components in a graph, usually when the graph is a grid.
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- [CSES Counting Rooms](https://cses.fi/problemset/task/1192)
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- [CSES Labyrinth](https://cses.fi/problemset/task/1193)
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### Tutorial
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@ -58,6 +64,8 @@ Author: Siyong Huang
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*Graph two-colorings* is assigning a boolean value to each node of the graph, dictated by the edge configuration
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The most common example of a two-colored graph is a *bipartite graph*, in which each edge connects two nodes of opposite colors
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- [CSES Building Teams](https://cses.fi/problemset/task/1668)
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### Tutorial
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The idea is that we can arbitrarily label a node and then run DFS. Every time we visit a new (unvisited) node, we set its color based on the edge rule. When we visit a previously visited node, check to see whether its color matches the edge rule. For example, an implementation of coloring a bipartite graph is shown below.
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@ -83,7 +91,7 @@ void dfs(int node)
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- Additional:
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- [Bipartite Graphs: cp-alg bipartite check](https://cp-algorithms.com/graph/bipartite-check.html)
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- Note: CP Algorithm uses bfs, but dfs accomplishes the same task
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- Note: CP Algorithm uses BFS, but DFS accomplishes the same task
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### Problems
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@ -91,12 +99,20 @@ void dfs(int node)
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## Cycle Detection
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A *cycle* is a non-empty path of distinct edges that start and end at the same node.
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*Cycle detection* determines properties of cycles in a graph, such as counting the number of cycles in a graph or determining whether a node is in a cycle. For most silver-level cycle problems, each node has only one out-degree, meaning that it's adjacency list is of size 1. If this is not the case, the problem generalizes to *Strongly Connected Components*, a platinum level concept.
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A *cycle* is a non-empty path of distinct edges that start and end at the same node. *Cycle detection* determines properties of cycles in a directed or undirected graph, such as whether each node of the graph is part of a cycle.
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### Tutorial
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### Functional Graphs
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The following sample code counts the number of cycles in a graph where each node points to one other node. The "stack" contains nodes that can reach the current node. If the current node points to a node v on the stack (on_stack[v] is true), then we know that a cycle has been created. However, if the current node points to a node v that has been previously visited but is not on the stack, then we know that the current chain of nodes points into a cycle that has already been considered.
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Links:
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* CPH 16.3: successor paths
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* CPH 16.4: cycle detection in successor graph
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In silver-level directed cycle problems, it is generally the case that each node has exactly one edge going out of it. This is known as a **successor graph** or a **functional graph.**
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The following sample code counts the number of cycles in such a graph. The "stack" contains nodes that can reach the current node. If the current node points to a node `v` on the stack (`on_stack[v]` is true), then we know that a cycle has been created. However, if the current node points to a node `v` that has been previously visited but is not on the stack, then we know that the current chain of nodes points into a cycle that has already been considered.
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(test code?)
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```cpp
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//Each node points to next_node[node]
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@ -122,10 +138,16 @@ int main()
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}
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```
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For non-functional directed graphs, this code will still detect a cycle if it exists (though `number_of_cycles` will be meaningless).
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### Problems
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- [Codeforces 1020B. Badge (Very Easy)](https://codeforces.com/contest/1020/problem/B)
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- Try to solve the problem in O(N)!
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- [The Bovine Shuffle (Normal)](http://usaco.org/index.php?page=viewproblem2&cpid=764)
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- [Swapity Swapity Swap (Very Hard)](http://www.usaco.org/index.php?page=viewproblem2&cpid=1014)
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- [CSES Round Trip (undirected)](https://cses.fi/problemset/task/1669)
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- [CSES Round Trip II (directed)](https://cses.fi/problemset/task/1678)
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Cycle finding is also related to **strongly connected components**, a platinum level concept.
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@ -70,5 +70,6 @@ Hasn't appeared in recent USACO Gold as far as I know.
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* [CSES High Score](https://cses.fi/problemset/task/1673)
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* [Kattis SSSP Negative](https://open.kattis.com/problems/shortestpath3)
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* [CSES Cycle Finding](https://cses.fi/problemset/task/1197)
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Can also modify Dijkstra's so it works with negative edge weights (but not negative cycles). The same running time bound no longer applies.
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@ -16,7 +16,7 @@ From the latter link (modified):
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> * It can be divided into multiple sections, where each section is a linear function (usually) with an integer slope.
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> * It is a convex/concave function. In other words, the slope of each section is non-decreasing or non-increasing when scanning the function from left to right.
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It's generally applicable as a DP optimization.
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It's generally applicable as a DP optimization. Usually you can come up with a slower DP (ex. $O(N^2)$) first and then optimize it to $O(N\log N)$ with slope trick.
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This document assumes some familiarity with at least one of the links above.
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@ -24,7 +24,7 @@ This document assumes some familiarity with at least one of the links above.
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[CF Buy Low Sell High](https://codeforces.com/contest/866/problem/D)
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Let $dp[i][j]$ denote the maximum amount of money you can have on day $i$ if you have exactly $j$ shares of stock on that day. The final answer will be $dp[N][0]$. This easily leads to an $O(N^2)$ DP.
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**Slow Solution**: Let $dp[i][j]$ denote the maximum amount of money you can have on day $i$ if you have exactly $j$ shares of stock on that day. The final answer will be $dp[N][0]$. This easily leads to an $O(N^2)$ DP.
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Of course, we never used the fact that the DP is concave down! Specifically, let $dif[i][j]=dp[i][j]-dp[i][j+1]\ge 0$. Then $dif[i][j]\le dif[i][j+1]$ for all $j\ge 0$ (ignoring the case when we get $dp$ values of $-\infty$).
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@ -69,9 +69,9 @@ int main() {
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Let $dif_i=a_i-b_i$. Defining $d_j=\sum_{i=1}^jdif_i$, our goal is to move around the potatoes such that $d_0,d_1,\ldots,d_N$ is a non-decreasing sequence. Moving a potato is equivalent to changing exactly one of the $d_i$ (aside from $d_0,d_N$) by one.
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As before, we can come up with a $O(N\cdot d_N)$ solution, where $dp[i][j]$ is the minimum cost to determine $d_0,d_1,\ldots,d_i$ such that $d_i\le j$. As before, this DP is concave up for a fixed $i$!!
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**Slow Solution:** Let $dp[i][j]$ be the minimum cost to determine $d_0,d_1,\ldots,d_i$ such that $d_i\le j$ for each $0\le j\le d_N$. This gives a $O(N\cdot d_N)$ solution.
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So given a piecewise linear function $DP_x$, we need to support the following operations.
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As before, this DP is concave up for a fixed $i$! Given a piecewise linear function $DP_x$, we need to support the following operations.
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* Add $|x-k|$ to the function for some $k$
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* Set $DP_x=\min(DP_x,DP_{x-1})$ for all $x$
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@ -181,11 +181,11 @@ int main() {
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## Problems
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* [Moving Haybales (USACO Camp)](https://probgate.org/viewproblem.php?pid=247)
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* [Wall](https://atcoder.jp/contests/kupc2016/tasks/kupc2016_h)
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* same as potatoes
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* [Stock Trading](https://probgate.org/viewproblem.php?pid=531&cid=81)
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* extension of buy low sell high
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* USACO Camp (private)
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* same as "Potatoes"
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* [Stock Trading (USACO Camp)](https://probgate.org/viewproblem.php?pid=531&cid=81)
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* extension of "Buy Low Sell High"
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* [Bookface](https://codeforces.com/group/ZFgXbZSjvp/contest/274852/problem/C)
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* [CCDSAP Exam](https://www.codechef.com/problems/CCDSAP)
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* [Farm of Monsters](https://codeforces.com/gym/102538/problem/F)
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- Standard Containers
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- ex. [Convention II](http://usaco.org/index.php?page=viewproblem2&cpid=859)
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- Sorting
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- Binary Search (BinSearch)
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- ex. [Counting Haybales](http://usaco.org/index.php?page=viewproblem2&cpid=666)
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- ex. [Counting Haybales](http://usaco.org/index.php?page=viewproblem2&cpid=666)
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- Binary Search (BinSearch)
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- ex. [Convention](http://usaco.org/index.php?page=viewproblem2&cpid=858)
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- Two Pointers (2P)
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- ex. [Diamond Collector](http://usaco.org/index.php?page=viewproblem2&cpid=643)
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- Prefix Sums (Psum)
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- ex. [Breed Counting](http://www.usaco.org/index.php?page=viewproblem2&cpid=572)
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- Flood Fill (FF)
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- DFS
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- ex. [Switching on the Lights](http://www.usaco.org/index.php?page=viewproblem2&cpid=570)
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- Occasionally, graph problems appear in bronze ...
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- Permutations
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- Cycle Decomposition?
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- ex. [Swapity Swapity Swap](http://www.usaco.org/index.php?page=viewproblem2&cpid=1014)
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# Gold
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- Dynamic Programming (DP)
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- ex. [Time is Mooney](http://www.usaco.org/index.php?page=viewproblem2&cpid=993)
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- Graphs
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- Shortest Path
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- Breadth First Search (BFS)
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- ex. [Cow Navigation](http://www.usaco.org/index.php?page=viewproblem2&cpid=695)
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- Shortest Path (SP)
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- ex. [Milk Pumping](http://www.usaco.org/index.php?page=viewproblem2&cpid=969)
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- Minimum Spanning Tree (MST)
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- Disjoint Set Union (DSU)
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- ex. [Fencedin](http://www.usaco.org/index.php?page=viewproblem2&cpid=623)
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- Topological Sort (TopoSort)
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- ex. [Timeline](http://www.usaco.org/index.php?page=viewproblem2&cpid=1017)
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- Minimum Spanning Tree (MST) & Disjoint Set Union (DSU)
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- ex. [Fencedin](http://www.usaco.org/index.php?page=viewproblem2&cpid=623)
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- Topological Sort (TopoSort)
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- ex. [Timeline](http://www.usaco.org/index.php?page=viewproblem2&cpid=1017)
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- [Range Sum Queries with Point Updates](https://thecodingwizard.github.io/usaco-training-2.0/Gold_1DRQ)
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- ex. [Haircut](http://www.usaco.org/index.php?page=viewproblem2&cpid=1041)
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- Number Theory
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- ex. [Cow Poetry](http://usaco.org/index.php?page=viewproblem2&cpid=897)
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- Less Common
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- Strings
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- Hashing
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- ex. [Cownomics](http://www.usaco.org/index.php?page=viewproblem2&cpid=741)
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- Combinatorics
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- Principle of Inclusion and Exclusion (PIE)
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- ex. [Cowpatibility](http://usaco.org/index.php?page=viewproblem2&cpid=862)
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- Sliding Window
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- ex. [Haybale Feast](http://www.usaco.org/index.php?page=viewproblem2&cpid=767)
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- Hashing (NOT COMMON)
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- ex. [Cownomics](http://www.usaco.org/index.php?page=viewproblem2&cpid=741)
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- Combinatorics (NOT COMMON)
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- Principle of Inclusion and Exclusion (PIE)
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- ex. [Cowpatibility](http://usaco.org/index.php?page=viewproblem2&cpid=862)
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- Sliding Window (NOT COMMON)
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- ex. [Haybale Feast](http://www.usaco.org/index.php?page=viewproblem2&cpid=767)
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# Platinum
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