Binary Search on Answer Technique in DSA C++ - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the main idea behind the Binary Search on Answer Technique?

It is a method to find an unknown value by guessing an answer range and using binary search to narrow down the correct answer based on a condition check.

beginner

In Binary Search on Answer, what does the 'answer' usually represent?

The 'answer' is the value we want to find, often a number like minimum or maximum that satisfies a given problem condition.

intermediate

Why do we use a condition function in Binary Search on Answer?

The condition function checks if a guessed answer is valid or not, helping us decide whether to search higher or lower in the answer range.

intermediate

How do you choose the initial search range in Binary Search on Answer?

You pick the smallest and largest possible values the answer can take, based on problem constraints or logical limits.

beginner

What is a real-life example where Binary Search on Answer can be applied?

Finding the minimum time needed to complete tasks when you can check if a given time is enough to finish all tasks.

What does Binary Search on Answer technique help you find?

A The middle element of an array

B An unknown value by narrowing down a range

C The sum of all elements in a list

D The maximum element in a sorted array

Binary Search on Answer narrows down the possible answer range to find the correct value.

Which of these is essential for Binary Search on Answer to work?

A Having a linked list

B Sorting the input array

C A condition function to check guesses

D Using recursion only

The condition function tells if the guessed answer is valid, guiding the search direction.

If the condition function returns true for a guess, what should you do next?

A Search for a smaller answer

B Search for a larger answer

C Stop immediately

D Ignore and guess randomly

If the guess is valid, try to find a smaller valid answer to optimize.

What is the time complexity of Binary Search on Answer if the condition check is O(n)?

A O(n^2)

B O(n)

C O(log n)

D O(n log m), where m is the answer range

Binary search runs in log m steps, each step doing O(n) work for condition check.

Which problem is a good candidate for Binary Search on Answer?

A Finding minimum maximum load to split tasks

B Sorting an array

C Finding duplicates in a list

D Reversing a string

Problems involving finding minimum or maximum values that satisfy conditions fit this technique.

Explain how Binary Search on Answer works step-by-step.

Think about guessing and checking answers like narrowing down a number.

Describe a problem scenario where Binary Search on Answer is useful and how you would apply it.

Consider tasks with time limits or capacity constraints.

Practice

1. What is the main idea behind the Binary Search on Answer technique?

easy

A. Searching for the answer by repeatedly dividing the search space based on feasibility

B. Sorting the array before applying binary search

C. Using binary search to find an element in a sorted array

D. Applying binary search only on arrays with unique elements

Solution

1. Step 1: Understand the problem context

   Binary Search on Answer is used when the answer lies within a range and we can check feasibility for a candidate answer.

2. Step 2: Apply binary search on the answer range

   We repeatedly pick a middle value and check if it satisfies the problem's conditions, narrowing the search space accordingly.

3. Final Answer:

   Searching for the answer by repeatedly dividing the search space based on feasibility -> Option A

4. Quick Check:

   Binary Search on Answer = feasibility check [OK]

Hint: Think of binary search on answer as guessing and checking [OK]

Common Mistakes:

• Confusing it with normal binary search on arrays
• Not checking feasibility properly
• Trying to apply it on unsorted data without a valid range

2. Which of the following is the correct syntax to start a binary search on answer for an integer range low to high in C++?

easy

A. while (low <= high) { int mid = (low + high) / 2; }

B. for (int mid = low; mid <= high; mid++) { }

C. while (low < high) { int mid = low + (high - low) / 2; }

D. if (low < high) { mid = (low + high) / 2; }

Solution

1. Step 1: Identify the correct loop condition

   Binary search typically uses a loop with condition low < high to avoid infinite loops and handle edge cases.

2. Step 2: Calculate mid safely

   Using mid = low + (high - low) / 2 prevents integer overflow compared to (low + high) / 2.

3. Final Answer:

   while (low < high) { int mid = low + (high - low) / 2; } -> Option C

4. Quick Check:

   Safe mid calculation with proper loop [OK]

Hint: Use low + (high - low)/2 to avoid overflow [OK]

Common Mistakes:

• Using <= in loop causing infinite loops
• Calculating mid as (low + high)/2 risking overflow
• Using for loop instead of while for binary search

3. Consider the following code snippet that uses binary search on answer to find the minimum maximum load for dividing an array into k subarrays:

bool canDivide(const vector<int>& arr, int k, int maxLoad) {
    int currentSum = 0, count = 1;
    for (int num : arr) {
        if (num > maxLoad) return false;
        if (currentSum + num > maxLoad) {
            count++;
            currentSum = num;
        } else {
            currentSum += num;
        }
    }
    return count <= k;
}

int binarySearchAnswer(const vector<int>& arr, int k) {
    int low = 0, high = 0;
    for (int num : arr) high += num;
    while (low < high) {
        int mid = low + (high - low) / 2;
        if (canDivide(arr, k, mid)) high = mid;
        else low = mid + 1;
    }
    return low;
}

int main() {
    vector<int> arr = {7, 2, 5, 10, 8};
    int k = 2;
    cout << binarySearchAnswer(arr, k) << endl;
}

What is the output of this program?

medium

A. 15

B. 18

C. 20

D. 22

Solution

1. Step 1: Understand the problem and code

   The code finds the minimum maximum sum when dividing the array into 2 subarrays. The total sum is 7+2+5+10+8=32.

2. Step 2: Trace binary search on answer

   Check mid values and feasibility:
   - mid=16: canDivide returns true (possible, e.g., subarrays sums 14 and 18)
   - mid=15: canDivide returns true (possible, e.g., subarrays sums 14 and 15)
   - mid=14: canDivide returns false (cannot split into 2 with max sum 14)
   Binary search narrows down to 15 as minimum max load.

3. Final Answer:

   15 -> Option A

4. Quick Check:

   Minimum max load = 15 [OK]

Hint: Sum array, binary search between 0 and sum [OK]

Common Mistakes:

• Not checking if single element > mid
• Incorrectly updating low and high
• Misunderstanding feasibility function

4. The following code attempts to find the smallest maximum weight capacity to ship packages within D days using binary search on answer. Identify the bug:

bool canShip(const vector<int>& weights, int D, int capacity) {
    int days = 1, currentLoad = 0;
    for (int w : weights) {
        if (w > capacity) return false;
        if (currentLoad + w > capacity) {
            days++;
            currentLoad = w;
        } else {
            currentLoad += w;
        }
    }
    return days <= D;
}

int shipWithinDays(const vector<int>& weights, int D) {
    int low = 0, high = accumulate(weights.begin(), weights.end(), 0);
    while (low <= high) {
        int mid = low + (high - low) / 2;
        if (canShip(weights, D, mid)) high = mid - 1;
        else low = mid + 1;
    }
    return low;
}

medium

A. The loop condition should be low < high instead of low <= high

B. The mid calculation can cause integer overflow

C. The canShip function does not handle empty weights array

D. The initial value of low should be the maximum weight in the array, not 0

Solution

1. Step 1: Analyze initial search range

   Low should start at the maximum single package weight because capacity cannot be less than the heaviest package.

2. Step 2: Identify impact of wrong low value

   Starting low at 0 causes unnecessary checks and may cause incorrect results or longer runtime.

3. Final Answer:

   The initial value of low should be the maximum weight in the array, not 0 -> Option D

4. Quick Check:

   Low = max(weights) [OK]

Hint: Low must be max element, high sum of all [OK]

Common Mistakes:

• Starting low at 0 instead of max element
• Using <= in loop causing off-by-one errors
• Ignoring edge cases with large weights

5. You want to find the minimum time required to complete n tasks using m machines, where each machine can complete one task in a fixed time given by array times. You can assign multiple tasks to a machine but tasks are done sequentially. Using binary search on answer, which approach correctly finds the minimum time to finish all tasks?

Choose the correct approach:

hard

A. Binary search on number of machines from 1 to m, checking if tasks can be done in fixed time

B. Binary search on time range from 0 to max(times) * n, and check if total tasks done by all machines in mid time is at least n

C. Sort times and pick the machine with minimum time multiplied by n

D. Use binary search on tasks from 1 to n, checking if tasks fit in given time

Solution

1. Step 1: Define search space for binary search

   The minimum time is 0 and maximum time is max machine time * n tasks (worst case all tasks on slowest machine).

2. Step 2: Check feasibility at mid time

   Calculate total tasks all machines can complete in mid time by summing floor(mid / times[i]). If total tasks >= n, mid time is feasible.

3. Step 3: Narrow search space

   If feasible, try smaller time by setting high = mid; else low = mid + 1.

4. Final Answer:

   Binary search on time range from 0 to max(times) * n, and check if total tasks done by all machines in mid time is at least n -> Option B

5. Quick Check:

   Binary search on time with feasibility check [OK]

Hint: Search time range, count tasks done at mid [OK]

Common Mistakes:

• Binary searching on machines instead of time
• Not summing tasks correctly for feasibility
• Ignoring sequential task processing per machine