ICPC 2014 - D. Game Strategy
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
Source-first archive entry
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competitive_programming/icpc/2014/D-game-strategy/solution.tex to update the written solution and
competitive_programming/icpc/2014/D-game-strategy/solution.cpp to update the implementation.
The website does not replace those files with hand-maintained HTML. It reads the copied source tree during the build and exposes the exact files below.
Problem Statement
Copied statement text kept beside the solution archive for direct reference.
Problem D
Game Strategy
Time Limit: 8 seconds
Alice and Bob are playing a board game. The board is divided into positions labeled a, b, c, d, . . . and
the players use a gamepiece to mark the current position. Each round of the game consists of two steps:
1. Alice makes a choice. Depending on the current position, she has different options, where each
option is a set of positions. Alice chooses one set S among the available sets of positions.
2. Bob makes a choice. His choice is one position p from the set S that Alice chose in step 1. Bob
moves the gamepiece to position p, which is the position for the start of the next round.
Prior to the first round, each player independently selects one of the positions and reveals it at the start
of the game. Bob’s position is where the game starts. Alice wins the game if she can force Bob to move
the gamepiece to the position she has chosen. To make things interesting, they have decided that Bob
will pay Alice a certain amount if he loses, but Alice must pay Bob a certain amount after every round.
The game now ends if Alice’s position is reached or when Alice runs out of cash.
Both Alice and Bob play optimally: Alice will always choose an option that will lead to her winning the
game, if this is possible, and Bob will always try to prevent Alice from winning.
For all possible start and end positions, Alice would like you to determine whether she can win the game
and if so, how many rounds it will take.
Input
The input consists of a single test case. The first line contains the number of positions n (1 ≤ n ≤ 25).
The n positions are labeled using the first n letters of the English alphabet in lowercase. The rest of the
test case consists of n lines, one for each position p, in alphabetical order. The line for position p contains
the options available to Alice in position p. It starts with the number of options m (1 ≤ m < 2n ), which
is followed by m distinct strings, one for each option. Each string contains the positions available to
Bob if Alice chooses that option. The string has at least 1 character, the characters (which correspond
to valid board positions) are in alphabetical order, and no characters are duplicated. The total number of
options for the test case is at most 106 .
Output
For each position p in alphabetical order, display one line. In that line, for each position q in alphabetical
order display the minimal number of rounds in which Alice can be guaranteed to arrive at position q
when starting the game in position p, or −1 if Alice cannot be guaranteed to reach q from p.
Sample Input 1 Sample Output 1
2 0 1
2 ab b -1 0
1 b
Sample Input 2 Sample Output 2
3 0 1 -1
1 b 1 0 -1
2 b a 2 2 0
2 ab ac
Editorial
Rendered from the copied solution.tex file. The original TeX source remains
available below.
Key Observations
Write the structural observations that make the problem tractable.
State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.
If the constraints matter, explain exactly which part of the solution they enable.
Algorithm
Describe the data structures and the state maintained by the algorithm.
Explain the processing order and why it is sufficient.
Mention corner cases explicitly if they affect the implementation.
Correctness Proof
We prove that the algorithm returns the correct answer.
Lemma 1.
State the first key claim.
Proof.
Provide a concise proof.
Lemma 2.
State the next claim if needed.
Proof.
Provide a concise proof.
Theorem.
The algorithm outputs the correct answer for every valid input.
Proof.
Combine the lemmas and finish the argument.
Complexity Analysis
State the running time and memory usage in terms of the input size.
Implementation Notes
Mention any non-obvious implementation detail that is easy to get wrong.
Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.
Code
Exact copied C++ implementation from solution.cpp.
#include <bits/stdc++.h>
using namespace std;
namespace {
void solve() {
// Fill in the full solution logic for the problem here.
}
} // namespace
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
solve();
return 0;
}
Source Files
Exact copied source-of-truth files. Edit solution.tex for the write-up and solution.cpp for the implementation.
\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{enumitem}
\title{ICPC World Finals 2014\\D. Game Strategy Time Limit: 8 seconds}
\author{}
\date{}
\begin{document}
\maketitle
\section*{Problem Summary}
State the problem in your own words. Focus on the mathematical or algorithmic core rather than repeating the full statement.
\section*{Key Observations}
\begin{itemize}[leftmargin=*]
\item Write the structural observations that make the problem tractable.
\item State any useful invariant, monotonicity property, graph interpretation, or combinatorial reformulation.
\item If the constraints matter, explain exactly which part of the solution they enable.
\end{itemize}
\section*{Algorithm}
\begin{enumerate}[leftmargin=*]
\item Describe the data structures and the state maintained by the algorithm.
\item Explain the processing order and why it is sufficient.
\item Mention corner cases explicitly if they affect the implementation.
\end{enumerate}
\section*{Correctness Proof}
We prove that the algorithm returns the correct answer.
\paragraph{Lemma 1.}
State the first key claim.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Lemma 2.}
State the next claim if needed.
\paragraph{Proof.}
Provide a concise proof.
\paragraph{Theorem.}
The algorithm outputs the correct answer for every valid input.
\paragraph{Proof.}
Combine the lemmas and finish the argument.
\section*{Complexity Analysis}
State the running time and memory usage in terms of the input size.
\section*{Implementation Notes}
\begin{itemize}[leftmargin=*]
\item Mention any non-obvious implementation detail that is easy to get wrong.
\item Mention numeric limits, indexing conventions, or tie-breaking rules if relevant.
\end{itemize}
\end{document}
#include <bits/stdc++.h>
using namespace std;
namespace {
void solve() {
// Fill in the full solution logic for the problem here.
}
} // namespace
int main() {
ios::sync_with_stdio(false);
cin.tie(nullptr);
solve();
return 0;
}