Adding the solutions

coin-change-2.java

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+
+// Solution -1
+
+// TimeComplexity: O(n × amount)
+// SpaceComplexity: O(amount)
+//Explanation: I use a 1D array arr where arr[j] represents the number of ways to make amount j. 
+// I initialize arr[0] = 1 because there is one way to make amount 0 (choose nothing). For each coin, I iterate through amounts and update
+class Solution {
+    public int change(int amount, int[] coins) {
+        int[] arr = new int[amount+1];
+        for (int j=0; j<amount+1; j++) {
+            if(j==0){
+                arr[j] =1;
+            } else{
+                arr[j] =0;
+            } 
+        }
+
+        for(int i=0; i<coins.length; i++){
+            for(int j=0; j<amount+1; j++){
+                if(j>=coins[i]) {
+                    arr[j] = arr[j] + arr[j-coins[i]];
+                }
+
+            }
+        }
+        return arr[amount];
+    }
+}
+
+
+// Solution - 2
+
+// TimeComplexity: O(n × amount)
+// SpaceComplexity: O(n × amount)
+// Explanation: Here dp[i][j] represents the number of ways to make amount j using the first i coins. For each coin, I either: Don’t choose it → dp[i-1][j]; Choose it → dp[i][j - coin]
+// I add both cases to get total combinations.
+
+
+class Solution {
+    public int change(int amount, int[] coins) {
+        int[][] dp = new int[coins.length+1][amount+1];
+        for(int i=1; i<coins.length+1; i++) {
+            for (int j=0; j<amount+1; j++){
+                if(j==0) {
+                    dp[i][j] =1;
+                } else if(j<coins[i-1]) {
+                    dp[i][j] = dp[i-1][j];
+                } else {
+                    dp[i][j] = dp[i-1][j] + dp[i][j-coins[i-1]];
+                }
+            }
+        } 
+        return dp[coins.length][amount]; 
+    }
+}
+
+
+// Solution - 3
+
+// TimeComplexity: O(n × amount)
+// SpaceComplexity: O(n × amount)
+// Explanation: I use recursion where at each step I either: Skip the current coin (move to next index)
+// Take the current coin (reduce amount but stay at same index). I store results in dp[idx][amount] to avoid recomputation. Each state is solved once
+
+class Solution {
+    public int change(int amount, int[] coins) {
+        Integer[][] dp = new Integer[coins.length+1][amount+1];
+        return helper(amount, coins, 0, dp);     
+    }
+
+    private int helper(int amount, int[] coins, int idx, Integer[][] dp){
+
+        // base
+        if(amount<0|| idx>=coins.length) return 0;
+        if(amount ==0) {
+            return 1;
+        }
+
+        if(dp[idx][amount] !=null) {
+            return dp[idx][amount];
+        }
+
+
+        // no choose
+        int case1= helper(amount, coins, idx+1,dp);
+
+        // choose
+        int case2= helper(amount-coins[idx], coins, idx, dp);
+        int sum = case1+case2;
+        dp[idx][amount] = sum;
+
+        return sum;
+    }
+}
+
+
+// Solution - 4
+
+
+// Time Complexity: O(2^n+m) where m is amount and n is length of coins array
+// Space Complexity: O(m+n)
+// Explanation: Here I recursively try both choices (take or skip coin) without memoization. 
+// This causes repeated recomputation of the same states, leading to exponential time complexity.
+
+class Solution {
+    public int change(int amount, int[] coins) {
+        return helper(amount, coins, 0);  
+
+    }
+
+    private int helper(int amount, int[] coins, int idx){
+        // base
+        if(amount<0|| idx>=coins.length) return 0;
+        if(amount ==0) {
+            return 1;
+        }
+
+        // no choose
+        int case1= helper(amount, coins, idx+1);
+
+        // choose
+        int case2= helper(amount-coins[idx], coins, idx);
+
+        return case1+case2;
+    }
+}

paint-house.java

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+// Solution - 1
+
+// TimeComplexity: O(n)
+// SpaceComplexity: O(1)
+// Explanation: I solve it from the last house to the first. The array dp[0], dp[1], dp[2] represents the minimum cost to paint from the next house onward
+//  if the current house is painted Red, Blue, or Green. For each house, I add its painting cost plus the minimum of the other two colors from the next house (since adjacent houses cannot have the same color).
+//  Finally, I return the minimum of the three values.
+
+class Solution {
+    public int minCost(int[][] costs) {
+        if(costs.length ==1) {
+            return Math.min(costs[0][0], Math.min(costs[0][1], costs[0][2]));
+        }
+        Integer[] dp = new Integer[3];
+        dp[0] = costs[costs.length-1][0];
+        dp[1] = costs[costs.length-1][1];
+        dp[2] = costs[costs.length-1][2];
+        for(int i =costs.length-2; i>-1; i--) {
+            int prev0 = dp[0];
+            int prev1 = dp[1];
+            int prev2 = dp[2];
+
+            dp[0] = costs[i][0] + Math.min(prev1, prev2);
+            dp[1] = costs[i][1] + Math.min(prev0, prev2);
+            dp[2] = costs[i][2] + Math.min(prev0, prev1);
+        }
+        return Math.min(dp[0], Math.min(dp[1], dp[2]));
+    }
+}
+
+
+// Solution - 2
+
+// TimeComplexity: O(n)
+// SpaceComplexity: O(n)
+// Explanation: Here I use a dp[i][color] table where it stores the minimum cost to paint from house i to the end if house i is painted with color.
+//  I fill the table from bottom to top using the same transition rule (choose the minimum of the other two colors from the next house). 
+// The answer is the minimum among the three colors for house 0.
+
+class Solution {
+    public int minCost(int[][] costs) {
+        if(costs.length ==1) {
+            return Math.min(costs[0][0], Math.min(costs[0][1], costs[0][2]));
+        }
+        Integer[][] dp = new Integer[costs.length][3];
+        dp[costs.length-1][0] = costs[costs.length-1][0];
+        dp[costs.length-1][1] = costs[costs.length-1][1];
+        dp[costs.length-1][2] = costs[costs.length-1][2];
+        for(int i =costs.length-2; i>-1; i--) {
+            dp[i][0] = costs[i][0] + Math.min(dp[i+1][1], dp[i+1][2]);
+            dp[i][1] = costs[i][1] + Math.min(dp[i+1][0], dp[i+1][2]);
+            dp[i][2] = costs[i][2] + Math.min(dp[i+1][0], dp[i+1][1]);
+        }
+        return Math.min(dp[0][0], Math.min(dp[0][1], dp[0][2]));
+    }
+}
+
+// Solution - 3
+
+// TimeComplexity: O(n)
+// SpaceComplexity: O(n)
+// Explanation: I use recursion and at each house decide the next house’s color (cannot be the same). 
+// The function returns the minimum cost starting from index idx with a given color. I store results in dp[idx][color] to avoid recomputation. 
+// This ensures each state is solved once.
+
+class Solution {
+    public int minCost(int[][] costs) {
+        Integer[][] dp = new Integer[costs.length][3];
+        // Red case
+        int case1= helper(costs, 0, 0, dp);
+        // blue case
+        int case2= helper(costs, 1, 0, dp);
+        // green case
+        int case3= helper(costs, 2, 0, dp);
+        return Math.min(case1, Math.min(case2, case3));
+    }
+
+    private int helper(int[][] costs, int color, int idx, Integer[][] dp) {
+        // base
+        if(idx==costs.length) return 0;
+        if(dp[idx][color]!=null) return dp[idx][color];
+
+        // logic
+        int case1;
+        int case2;
+        if(color==0) {
+            case1 = costs[idx][color] + helper(costs, 1, idx+1, dp);
+            case2 = costs[idx][color] + helper(costs, 2, idx+1, dp);
+
+        } else if(color==1) {
+            case1 = costs[idx][color] + helper(costs, 0, idx+1, dp);
+            case2 = costs[idx][color] + helper(costs, 2, idx+1, dp);
+        } else {
+            case1 = costs[idx][color] + helper(costs, 0, idx+1, dp);
+            case2 = costs[idx][color] + helper(costs, 1, idx+1, dp);
+        }
+        int min = Math.min(case1, case2);
+        dp[idx][color] = min;
+        return min;
+    }
+}
+
+
+
+// Solution - 4
+
+// TimeComplexity: O(2ⁿ)
+// SpaceComplexity: O(n)
+// Explanation: Here I recursively try both possible colors for the next house at every step without storing results. 
+// This causes repeated calculations of the same states, leading to exponential time complexity.
+
+
+class Solution {
+    public int minCost(int[][] costs) {
+        // Red case
+        int case1= helper(costs, 0, 0);
+        // blue case
+        int case2= helper(costs, 1, 0);
+        // green case
+        int case3= helper(costs, 2, 0);
+        return Math.min(case1, Math.min(case2, case3));
+    }
+
+    private int helper(int[][] costs, int color, int idx) {
+        // base
+        if(idx==costs.length) return 0;
+
+        // logic
+        int case1;
+        int case2;
+        if(color==0) {
+            case1 = costs[idx][color] + helper(costs, 1, idx+1);
+            case2 = costs[idx][color] + helper(costs, 2, idx+1);
+
+        } else if(color==1) {
+            case1 = costs[idx][color] + helper(costs, 0, idx+1);
+            case2 = costs[idx][color] + helper(costs, 2, idx+1);
+        } else {
+            case1 = costs[idx][color] + helper(costs, 0, idx+1);
+            case2 = costs[idx][color] + helper(costs, 1, idx+1);
+        }
+        return Math.min(case1, case2);
+    }
+}