花花酱 LeetCode 741. Cherry Pickup

By zxi on December 4, 2017

题目大意：给你樱桃田的地图（1: 樱桃, 0: 空, -1: 障碍物）。然你从左上角走到右下角（只能往右或往下），再从右下角走回左上角（只能往左或者往上）。问你最多能采到多少棵樱桃。

Problem:

In a N x N grid representing a field of cherries, each cell is one of three possible integers.

0 means the cell is empty, so you can pass through;
1 means the cell contains a cherry, that you can pick up and pass through;
-1 means the cell contains a thorn that blocks your way.

Your task is to collect maximum number of cherries possible by following the rules below:

Starting at the position (0, 0) and reaching (N-1, N-1) by moving right or down through valid path cells (cells with value 0 or 1);
After reaching (N-1, N-1), returning to (0, 0) by moving left or up through valid path cells;
When passing through a path cell containing a cherry, you pick it up and the cell becomes an empty cell (0);
If there is no valid path between (0, 0) and (N-1, N-1), then no cherries can be collected.

Example 1:

Input: grid =

[[0, 1, -1],

[1, 0, -1],

[1, 1, 1]]

Output: 5

Explanation:

The player started at (0, 0) and went down, down, right right to reach (2, 2).

4 cherries were picked up during this single trip, and the matrix becomes [[0,1,-1],[0,0,-1],[0,0,0]].

Then, the player went left, up, up, left to return home, picking up one more cherry.

The total number of cherries picked up is 5, and this is the maximum possible.

Note:

grid is an N by N 2D array, with 1 <= N <= 50.
Each grid[i][j] is an integer in the set {-1, 0, 1}.
It is guaranteed that grid[0][0] and grid[N-1][N-1] are not -1.

Idea:

Key observation: (0,0) to (n-1, n-1) to (0, 0) is the same as (n-1,n-1) to (0,0) twice

Two people starting from (n-1, n-1) and go to (0, 0).
They move one step (left or up) at a time simultaneously. And pick up the cherry within the grid (if there is one).
if they ended up at the same grid with a cherry. Only one of them can pick up it.

Solution: DP / Recursion with memoization.

x1, y1, x2 to represent a state y2 can be computed: y2 = x1 + y1 – x2

dp(x1, y1, x2) computes the max cherries if start from {(x1, y1), (x2, y2)} to (0, 0), which is a recursive function.

Since two people move independently, there are 4 subproblems: (left, left), (left, up), (up, left), (left, up). Finally, we have:

dp(x1, y1, x2)= g[y1][x1] + g[y2][x2] + max{dp(x1-1,y1,x2-1), dp(x1,y1-1,x2-1), dp(x1-1,y1,x2), dp(x1,y1-1,x2)}

Time complexity: O(n^3)

Space complexity: O(n^3)

Solution: DP

Time complexity: O(n^3)

Space complexity: O(n^3)

C ++

// Author: Huahua

// Runtime: 32 ms

class Solution {

public:

int cherryPickup(vector<vector<int>>& grid) {

const int n = grid.size();

grid_ = &grid;

m_ = vector<vector<vector<int>>>(

n, vector<vector<int>>(n, vector<int>(n, INT_MIN)));

return max(0, dp(n - 1, n - 1, n - 1));

}

private:

// max cherries from (x1, y1) to (0, 0) + (x2, y2) to (0, 0)

int dp(int x1, int y1, int x2) {

const int y2 = x1 + y1 - x2;

if (x1 < 0 || y1 < 0 || x2 < 0 || y2 < 0) return -1;

if ((*grid_)[y1][x1] < 0 || (*grid_)[y2][x2] < 0) return -1;

if (x1 == 0 && y1 == 0) return (*grid_)[y1][x1];

if (m_[x1][y1][x2] != INT_MIN) return m_[x1][y1][x2];

int ans = max(max(dp(x1 - 1, y1, x2 - 1), dp(x1, y1 - 1, x2)),

max(dp(x1, y1 - 1, x2 - 1), dp(x1 - 1, y1, x2)));

if (ans < 0) return m_[x1][y1][x2] = -1;

ans += (*grid_)[y1][x1];

if (x1 != x2) ans += (*grid_)[y2][x2];

return m_[x1][y1][x2] = ans;

}

vector<vector<vector<int>>> m_;

vector<vector<int>>* grid_; // does not own the object

};

Java

// Author: Huahua

class Solution {

private int[][] grid;

private int[][][] memo;

public int cherryPickup(int[][] grid) {

this.grid = grid;

int m = grid.length;

int n = grid[0].length;

memo = new int[m][n][n];

for (int i = 0; i < m; ++i)

for (int j = 0; j < n; ++j)

Arrays.fill(memo[i][j], Integer.MIN_VALUE);

return Math.max(0, dp(n - 1, m - 1, n - 1));

}

private int dp(int x1, int y1, int x2) {

final int y2 = x1 + y1 - x2;

if (x1 < 0 || y1 < 0 || x2 < 0 || y2 < 0) return -1;

if (grid[y1][x1] < 0 || grid[y2][x2] < 0) return -1;

if (x1 == 0 && y1 == 0) return grid[y1][x1];

if (memo[y1][x1][x2] != Integer.MIN_VALUE) return memo[y1][x1][x2];

memo[y1][x1][x2] = Math.max(Math.max(dp(x1 - 1, y1, x2 - 1), dp(x1, y1 - 1, x2)),

Math.max(dp(x1, y1 - 1, x2 - 1), dp(x1 - 1, y1, x2)));

if (memo[y1][x1][x2] >= 0) {

memo[y1][x1][x2] += grid[y1][x1];

if (x1 != x2) memo[y1][x1][x2] += grid[y2][x2];

}

return memo[y1][x1][x2];

}

Related Problems:

花花酱 LeetCode 735. Asteroid Collision

By zxi on December 3, 2017

Problem:

We are given an array asteroids of integers representing asteroids in a row.

For each asteroid, the absolute value represents its size, and the sign represents its direction (positive meaning right, negative meaning left). Each asteroid moves at the same speed.

Find out the state of the asteroids after all collisions. If two asteroids meet, the smaller one will explode. If both are the same size, both will explode. Two asteroids moving in the same direction will never meet.

Example 1:

Input:

asteroids = [5, 10, -5]

Output: [5, 10]

Explanation:

The 10 and -5 collide resulting in 10. The 5 and 10 never collide.

Example 2:

Input:

asteroids = [8, -8]

Output: []

Explanation:

The 8 and -8 collide exploding each other.

Example 3:

Input:

asteroids = [10, 2, -5]

Output: [10]

Explanation:

The 2 and -5 collide resulting in -5. The 10 and -5 collide resulting in 10.

Example 4:

Input:

asteroids = [-2, -1, 1, 2]

Output: [-2, -1, 1, 2]

Explanation:

The -2 and -1 are moving left, while the 1 and 2 are moving right.

Asteroids moving the same direction never meet, so no asteroids will meet each other.

Note:

The length of asteroids will be at most 10000.
Each asteroid will be a non-zero integer in the range [-1000, 1000]..

Idea:

Simulation

Time complexity: O(n), Space complexity: O(n)

Solution:

C ++

// Author: Huahua

// Runtime: 22 ms

class Solution {

public:

vector<int> asteroidCollision(vector<int>& asteroids) {

vector<int> s;

for (int i = 0 ; i < asteroids.size(); ++i) {

const int size = asteroids[i];

if (size > 0) { // To right, OK

s.push_back(size);

} else {

// To left

if (s.empty() || s.back() < 0) // OK when all are negtives

s.push_back(size);

else if (abs(s.back()) <= abs(size)) {

// Top of the stack is going right.

// Its size is less than the current one.

// The current one still alive!

if (abs(s.back()) < abs(size)) --i;

s.pop_back(); // Destory the top one moving right

}

// s must look like [-s1, -s2, ... , si, sj, ...]

return s;

}

};

花花酱 LeetCode 11. Container With Most Water

By zxi on December 3, 2017

Problem:

Given n non-negative integers a₁, a₂, …, a_n, where each represents a point at coordinate (i, a_i). n vertical lines are drawn such that the two endpoints of line i is at (i, a_i) and (i, 0). Find two lines, which together with x-axis forms a container, such that the container contains the most water.

Note: You may not slant the container and n is at least 2.

Examples:

input: [1 3 2 4]
output: 6
explanation: use 3, 4, we have the following, which contains (4th-2nd) * min(3, 4) = 2 * 3 = 6 unit of water.

|~~|

Idea:

Two pointers

Time complexity: O(n)

Space complexity: O(1)

Solution:

C++ two pointers

// Author: Huahua

// Runtime: 19 ms

class Solution {

public:

int maxArea(const vector<int>& height) {

int ans = 0;

int l = 0;

int r = height.size() - 1;

while (l < r) {

int h = min(height[l], height[r]);

ans = max(ans, h * (r - l));

if (height[l] < height[r])

++l;

else

--r;

}

return ans;

}

};

Java

// Author: Huahua

// Runtime: 13 ms

class Solution {

public int maxArea(int[] height) {

int l = 0;

int r = height.length - 1;

int ans = 0;

while (l < r) {

int h = Math.min(height[l], height[r]);

ans = Math.max(ans, h * (r - l));

if (height[l] < height[r])

++l;

else

--r;

}

return ans;

}

花花酱 LeetCode 399. Evaluate Division

By zxi on December 1, 2017

题目大意：给你一些含有变量名的分式的值，让你计算另外一些分式的值，如果不能计算返回-1。

Problem:

Equations are given in the format A / B = k, where A and B are variables represented as strings, and k is a real number (floating point number). Given some queries, return the answers. If the answer does not exist, return -1.0.

Example:
Given a / b = 2.0, b / c = 3.0.
queries are: a / c = ?, b / a = ?, a / e = ?, a / a = ?, x / x = ? .
return [6.0, 0.5, -1.0, 1.0, -1.0 ].

The input is: vector<pair<string, string>> equations, vector<double>& values, vector<pair<string, string>> queries , where equations.size() == values.size(), and the values are positive. This represents the equations. Return vector<double>.

According to the example above:

equations = [ ["a", "b"], ["b", "c"] ],

values = [2.0, 3.0],

queries = [ ["a", "c"], ["b", "a"], ["a", "e"], ["a", "a"], ["x", "x"] ].

The input is always valid. You may assume that evaluating the queries will result in no division by zero and there is no contradiction.

Solution 1: DFS

C++

// Author: Huahua

// Runtime: 3 ms

class Solution {

public:

vector<double> calcEquation(vector<pair<string, string>> equations, vector<double>& values, vector<pair<string, string>> queries) {

// g[A][B] = k -> A / B = k

unordered_map<string, unordered_map<string, double>> g;

for (int i = 0; i < equations.size(); ++i) {

const string& A = equations[i].first;

const string& B = equations[i].second;

const double k = values[i];

g[A][B] = k;

g[B][A] = 1.0 / k;

}

vector<double> ans;

for (const auto& pair : queries) {

const string& X = pair.first;

const string& Y = pair.second;

if (!g.count(X) || !g.count(Y)) {

ans.push_back(-1.0);

continue;

}

unordered_set<string> visited;

ans.push_back(divide(X, Y, g, visited));

}

return ans;

}

private:

// get result of A / B

double divide(const string& A, const string& B,

unordered_map<string, unordered_map<string, double>>& g,

unordered_set<string>& visited) {

if (A == B) return 1.0;

visited.insert(A);

for (const auto& pair : g[A]) {

const string& C = pair.first;

if (visited.count(C)) continue;

double d = divide(C, B, g, visited); // d = C / B

// A / B = C / B * A / C

if (d > 0) return d * g[A][C];

}

return -1.0;

}

};

Java

// Updated: 2020/9/27

// Author: Huahua

// Running time: 1 ms

class Solution {

private Map<String, HashMap<String, Double>> g = new HashMap<>();

public double[] calcEquation(List<List<String>> equations, double[] values, List<List<String>> queries) {

for (int i = 0; i < equations.size(); ++i) {

String x = equations.get(i).get(0);

String y = equations.get(i).get(1);

double k = values[i];

g.computeIfAbsent(x, l -> new HashMap<String, Double>()).put(y, k);

g.computeIfAbsent(y, l -> new HashMap<String, Double>()).put(x, 1.0 / k);

}

double[] ans = new double[queries.size()];

for (int i = 0; i < queries.size(); ++i) {

String x = queries.get(i).get(0);

String y = queries.get(i).get(1);

if (!g.containsKey(x) || !g.containsKey(y)) {

ans[i] = -1.0;

} else {

ans[i] = divide(x, y, new HashSet<String>());

}

return ans;

}

private double divide(String x, String y, Set<String> visited) {

if (x.equals(y)) return 1.0;

visited.add(x);

if (!g.containsKey(x)) return -1.0;

for (String n : g.get(x).keySet()) {

if (visited.contains(n)) continue;

visited.add(n);

double d = divide(n, y, visited);

if (d > 0) return d * g.get(x).get(n);

}

return -1.0;

}

Python3

"""

Author: Huahua

Running time: 32 ms (beats 100%)

"""

class Solution:

def calcEquation(self, equations, values, queries):

def divide(x, y, visited):

if x == y: return 1.0

visited.add(x)

for n in g[x]:

if n in visited: continue

visited.add(n)

d = divide(n, y, visited)

if d > 0: return d * g[x][n]

return -1.0

g = collections.defaultdict(dict)

for (x, y), v in zip(equations, values):

g[x][y] = v

g[y][x] = 1.0 / v

ans = [divide(x, y, set()) if x in g and y in g else -1 for x, y in queries]

return ans

Solution 2: Union Find

C++

// Author: Huahua

// Runtime: 3 ms

class Solution {

public:

vector<double> calcEquation(const vector<pair<string, string>>& equations, vector<double>& values, const vector<pair<string, string>>& queries) {

// parents["A"] = {"B", 2.0} -> A = 2.0 * B

// parents["B"] = {"C", 3.0} -> B = 3.0 * C

unordered_map<string, pair<string, double>> parents;

for (int i = 0; i < equations.size(); ++i) {

const string& A = equations[i].first;

const string& B = equations[i].second;

const double k = values[i];

// Neighter is in the forrest

if (!parents.count(A) && !parents.count(B)) {

parents[A] = {B, k};

parents[B] = {B, 1.0};

} else if (!parents.count(A)) {

parents[A] = {B, k};

} else if (!parents.count(B)) {

parents[B] = {A, 1.0 / k};

} else {

auto& rA = find(A, parents);

auto& rB = find(B, parents);

parents[rA.first] = {rB.first, k / rA.second * rB.second};

}

vector<double> ans;

for (const auto& pair : queries) {

const string& X = pair.first;

const string& Y = pair.second;

if (!parents.count(X) || !parents.count(Y)) {

ans.push_back(-1.0);

continue;

}

auto& rX = find(X, parents); // {rX, X / rX}

auto& rY = find(Y, parents); // {rY, Y / rY}

if (rX.first != rY.first)

ans.push_back(-1.0);

else // X / Y = (X / rX / (Y / rY))

ans.push_back(rX.second / rY.second);

}

return ans;

}

private:

pair<string, double>& find(const string& C, unordered_map<string, pair<string, double>>& parents) {

if (C != parents[C].first) {

const auto& p = find(parents[C].first, parents);

parents[C].first = p.first;

parents[C].second *= p.second;

}

return parents[C];

}

};

Java

// Updated: 9/27/2020

// Author: Huahua

// Running time: 3 ms

class Solution {

class Node {

public String parent;

public double ratio;

public Node(String parent, double ratio) {

this.parent = parent;

this.ratio = ratio;

}

class UnionFindSet {

private Map<String, Node> parents = new HashMap<>();

public Node find(String s) {

if (!parents.containsKey(s)) return null;

Node n = parents.get(s);

if (!n.parent.equals(s)) {

Node p = find(n.parent);

n.parent = p.parent;

n.ratio *= p.ratio;

}

return n;

}

public void union(String A, String B, double ratio) {

boolean hasA = parents.containsKey(A);

boolean hasB = parents.containsKey(B);

if (!hasA && !hasB) {

parents.put(A, new Node(B, ratio));

parents.put(B, new Node(B, 1.0));

} else if (!hasA) {

parents.put(A, new Node(B, ratio));

} else if (!hasB) {

parents.put(B, new Node(A, 1.0 / ratio));

} else {

Node rA = find(A);

Node rB = find(B);

parents.put(rA.parent,

new Node(rB.parent, ratio / rA.ratio * rB.ratio));

}

public double[] calcEquation(List<List<String>> equations, double[] values, List<List<String>> queries) {

UnionFindSet u = new UnionFindSet();

for (int i = 0; i < equations.size(); ++i)

u.union(equations.get(i).get(0), equations.get(i).get(1), values[i]);

double[] ans = new double[queries.size()];

for (int i = 0; i < queries.size(); ++i) {

Node rx = u.find(queries.get(i).get(0));

Node ry = u.find(queries.get(i).get(1));

if (rx == null || ry == null || !rx.parent.equals(ry.parent))

ans[i] = -1.0;

else

ans[i] = rx.ratio / ry.ratio;

}

return ans;

}

Python3

"""

Author: Huahua

Running time: 32 ms (beats 100%)

"""

class Solution:

def calcEquation(self, equations, values, queries):

def find(x):

if x != U[x][0]:

px, pv = find(U[x][0])

U[x] = (px, U[x][1] * pv)

return U[x]

def divide(x, y):

rx, vx = find(x)

ry, vy = find(y)

if rx != ry: return -1.0

return vx / vy

U = {}

for (x, y), v in zip(equations, values):

if x not in U and y not in U:

U[x] = (y, v)

U[y] = (y, 1.0)

elif x not in U:

U[x] = (y, v)

elif y not in U:

U[y] = (x, 1.0 / v)

else:

rx, vx = find(x)

ry, vy = find(y)

U[rx] = (ry, v / vx * vy)

ans = [divide(x, y) if x in U and y in U else -1 for x, y in queries]

return ans

Related Problems:

花花酱 LeetCode 736. Parse Lisp Expression

By zxi on December 1, 2017

Problem:

You are given a string expression representing a Lisp-like expression to return the integer value of.

The syntax for these expressions is given as follows.

An expression is either an integer, a let-expression, an add-expression, a mult-expression, or an assigned variable. Expressions always evaluate to a single integer.
(An integer could be positive or negative.)
A let-expression takes the form (let v1 e1 v2 e2 ... vn en expr), where let is always the string "let", then there are 1 or more pairs of alternating variables and expressions, meaning that the first variable v1is assigned the value of the expression e1, the second variable v2 is assigned the value of the expression e2, and so on sequentially; and then the value of this let-expression is the value of the expression expr.
An add-expression takes the form (add e1 e2) where add is always the string "add", there are always two expressions e1, e2, and this expression evaluates to the addition of the evaluation of e1 and the evaluation of e2.
A mult-expression takes the form (mult e1 e2) where mult is always the string "mult", there are always two expressions e1, e2, and this expression evaluates to the multiplication of the evaluation of e1 and the evaluation of e2.
For the purposes of this question, we will use a smaller subset of variable names. A variable starts with a lowercase letter, then zero or more lowercase letters or digits. Additionally for your convenience, the names “add”, “let”, or “mult” are protected and will never be used as variable names.
Finally, there is the concept of scope. When an expression of a variable name is evaluated, within the context of that evaluation, the innermost scope (in terms of parentheses) is checked first for the value of that variable, and then outer scopes are checked sequentially. It is guaranteed that every expression is legal. Please see the examples for more details on scope.

Evaluation Examples:

Input: (add 1 2)

Output: 3

Input: (mult 3 (add 2 3))

Output: 15

Input: (let x 2 (mult x 5))

Output: 10

Input: (let x 2 (mult x (let x 3 y 4 (add x y))))

Output: 14

Explanation: In the expression (add x y), when checking for the value of the variable x,

we check from the innermost scope to the outermost in the context of the variable we are trying to evaluate.

Since x = 3 is found first, the value of x is 3.

Input: (let x 3 x 2 x)

Output: 2

Explanation: Assignment in let statements is processed sequentially.

Input: (let x 1 y 2 x (add x y) (add x y))

Output: 5

Explanation: The first (add x y) evaluates as 3, and is assigned to x.

The second (add x y) evaluates as 3+2 = 5.

Input: (let x 2 (add (let x 3 (let x 4 x)) x))

Output: 6

Explanation: Even though (let x 4 x) has a deeper scope, it is outside the context

of the final x in the add-expression. That final x will equal 2.

Input: (let a1 3 b2 (add a1 1) b2)

Output 4

Explanation: Variable names can contain digits after the first character.

Note:

The given string expression is well formatted: There are no leading or trailing spaces, there is only a single space separating different components of the string, and no space between adjacent parentheses. The expression is guaranteed to be legal and evaluate to an integer.
The length of expression is at most 2000. (It is also non-empty, as that would not be a legal expression.)
The answer and all intermediate calculations of that answer are guaranteed to fit in a 32-bit integer.

Idea:

Recursive parsing

Time complexity: O(n^2) in worst case O(n) in practice

Space complexity: O(n)

// Author: Huahua

// Runtime: 3 ms

class Solution {

public:

int evaluate(string expression) {

scopes_.clear();

int pos = 0;

return eval(expression, pos);

}

private:

int eval(const string& s, int& pos) {

scopes_.push_front(unordered_map<string, int>());

int value = 0; // The return value of current expr

if (s[pos] == '(') ++pos;

// command, variable or number

const string token = getToken(s, pos);

if (token == "add") {

int v1 = eval(s, ++pos);

int v2 = eval(s, ++pos);

value = v1 + v2;

} else if (token == "mult") {

int v1 = eval(s, ++pos);

int v2 = eval(s, ++pos);

value = v1 * v2;

} else if (token == "let") {

string var;

// expecting " var1 exp1 var2 exp2 ... last_expr)"

while (s[pos] != ')') {

++pos;

// Must be last_expr

if (s[pos] == '(') {

value = eval(s, ++pos);

break;

}

// Get a token, could be "x" or "-12" for last_expr

var = getToken(s, pos);

// End of let, var is last_expr

if (s[pos] == ')') {

if (isalpha(var[0]))

value = getValue(var);

else

value = stoi(var);

break;

}

// x -12 -> set x to -12 and store in the current scope and take it as the current return value

value = scopes_.front()[var] = eval(s, ++pos);

}

} else if (isalpha(token[0])) {

value = getValue(token); // symbol

} else {

value = std::stoi(token); // number

}

if (s[pos] == ')') ++pos;

scopes_.pop_front();

return value;

}

int getValue(const string& symbol) {

for (const auto& scope : scopes_)

if (scope.count(symbol)) return scope.at(symbol);

return 0;

}

// Get a token from current pos.

// "let x" -> "let"

// "-12 (add x y)" -> "-12"

string getToken(const string& s, int& pos) {

string token;

while (pos < s.length()) {

if (s[pos] == ')' || s[pos] == ' ') break;

token += s[pos++];

}

return token;

}

deque<unordered_map<string, int>> scopes_;

};

Huahua's Tech Road

花花酱 LeetCode 741. Cherry Pickup

Solution: DP

C ++

Java

花花酱 LeetCode 735. Asteroid Collision

花花酱 LeetCode 11. Container With Most Water

花花酱 LeetCode 399. Evaluate Division

Solution 1: DFS

C++

Java

Python3

Solution 2: Union Find

C++

Java

Python3

花花酱 LeetCode 736. Parse Lisp Expression