Activity Selection Problem in DSA Typescript - Time & Space Complexity

Time Complexity: Activity Selection Problem

O(n)

Understanding Time Complexity

We want to know how the time needed to select activities grows as the number of activities increases.

How does the algorithm's work change when we have more activities to choose from?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function activitySelection(start: number[], finish: number[]): number[] {
  const n = start.length;
  const selected: number[] = [];
  let lastFinish = 0;
  for (let i = 0; i < n; i++) {
    if (start[i] >= lastFinish) {
      selected.push(i);
      lastFinish = finish[i];
    }
  }
  return selected;
}
    

This code picks the maximum number of activities that don't overlap, assuming activities are sorted by finish time.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: A single loop that goes through all activities once.
• How many times: Exactly once for each activity, so n times.

How Execution Grows With Input

As the number of activities grows, the work grows in a straight line.

Input Size (n)	Approx. Operations
10	About 10 checks
100	About 100 checks
1000	About 1000 checks

Pattern observation: The number of operations grows directly with the number of activities.

Final Time Complexity

Time Complexity: O(n)

This means the time to select activities grows in direct proportion to how many activities there are.

Common Mistake

[X] Wrong: "The algorithm checks every pair of activities, so it must be quadratic."

[OK] Correct: The code only loops once through the list, not nested loops, so it does not compare every pair.

Interview Connect

Understanding this linear time solution shows you can spot efficient ways to handle scheduling problems, a useful skill in many coding challenges.

Self-Check

"What if the activities were not sorted by finish time? How would the time complexity change?"

Practice

1. What is the main goal of the Activity Selection Problem?

easy

A. Select the maximum number of activities that don't overlap in time

B. Find the activity with the longest duration

C. Sort activities by their start times

D. Calculate the total time of all activities

Solution

1. Step 1: Understand the problem goal

   The Activity Selection Problem aims to choose activities that do not overlap and maximize the count.

2. Step 2: Identify the correct objective

   Among the options, only selecting the maximum number of non-overlapping activities matches the problem's goal.

3. Final Answer:

   Select the maximum number of activities that don't overlap in time -> Option A

4. Quick Check:

   Goal = Max non-overlapping activities [OK]

Hint: Focus on maximizing count without time conflicts [OK]

Common Mistakes:

• Confusing longest duration with maximum count
• Sorting by start time only
• Summing all activity durations

2. Which of the following TypeScript code snippets correctly sorts activities by their finish time for the Activity Selection Problem?

easy

A. activities.sort((a, b) => a.finish - b.finish);

B. activities.sort((a, b) => a.start - b.start);

C. activities.sort((a, b) => b.finish - a.finish);

D. activities.sort((a, b) => b.start - a.start);

Solution

1. Step 1: Identify sorting criteria for Activity Selection

   Activities must be sorted by their finish times in ascending order to apply the greedy approach.

2. Step 2: Match sorting code to criteria

   activities.sort((a, b) => a.finish - b.finish); sorts activities by finish time ascending, which is correct for the problem.

3. Final Answer:

   activities.sort((a, b) => a.finish - b.finish); -> Option A

4. Quick Check:

   Sort by finish time ascending [OK]

Hint: Sort activities by finish time ascending [OK]

Common Mistakes:

• Sorting by start time instead of finish time
• Sorting finish time descending
• Using incorrect comparison operators

3. What is the output of the following TypeScript code for the Activity Selection Problem?

const activities = [
  {start: 1, finish: 4},
  {start: 3, finish: 5},
  {start: 0, finish: 6},
  {start: 5, finish: 7},
  {start: 8, finish: 9},
  {start: 5, finish: 9}
];

activities.sort((a, b) => a.finish - b.finish);

const selected = [];
let lastFinish = 0;
for (const activity of activities) {
  if (activity.start >= lastFinish) {
    selected.push(activity);
    lastFinish = activity.finish;
  }
}

console.log(selected);

medium

A. [{start:0, finish:6}, {start:8, finish:9}]

B. [{start:1, finish:4}, {start:3, finish:5}, {start:5, finish:7}]

C. [{start:1, finish:4}, {start:5, finish:7}, {start:8, finish:9}]

D. [{start:3, finish:5}, {start:5, finish:9}]

Solution

1. Step 1: Sort activities by finish time

   Sorted list: [{1,4}, {3,5}, {0,6}, {5,7}, {8,9}, {5,9}]

2. Step 2: Select activities greedily

   Select {1,4} (start 1 >= 0), then {5,7} (start 5 >= 4), then {8,9} (start 8 >= 7). Others overlap.

3. Final Answer:

   [{start:1, finish:4}, {start:5, finish:7}, {start:8, finish:9}] -> Option C

4. Quick Check:

   Greedy selection = chosen activities [OK]

Hint: Pick earliest finishing activities first [OK]

Common Mistakes:

• Selecting overlapping activities
• Not sorting by finish time
• Updating lastFinish incorrectly

4. Identify the bug in this TypeScript code for the Activity Selection Problem:

const activities = [
  {start: 1, finish: 3},
  {start: 2, finish: 5},
  {start: 4, finish: 7}
];

activities.sort((a, b) => a.start - b.start);

const selected = [];
let lastFinish = 0;
for (const activity of activities) {
  if (activity.start >= lastFinish) {
    selected.push(activity);
    lastFinish = activity.finish;
  }
}

console.log(selected);

medium

A. Not initializing selected array

B. Sorting by start time instead of finish time

C. Using 'let' instead of 'const' for lastFinish

D. Incorrect comparison operator in if condition

Solution

1. Step 1: Check sorting criteria

   The code sorts activities by start time, but the greedy algorithm requires sorting by finish time.

2. Step 2: Identify impact of sorting error

   Sorting by start time can cause incorrect selection of overlapping activities, breaking the algorithm.

3. Final Answer:

   Sorting by start time instead of finish time -> Option B

4. Quick Check:

   Sort by finish time needed [OK]

Hint: Always sort by finish time for this problem [OK]

Common Mistakes:

• Sorting by start time
• Confusing variable declarations
• Wrong if condition operators

5. Given the activities with start and finish times:

const activities = [
  {start: 0, finish: 3},
  {start: 1, finish: 4},
  {start: 3, finish: 5},
  {start: 4, finish: 7},
  {start: 5, finish: 9},
  {start: 8, finish: 10}
];

How many activities can be selected using the Activity Selection Problem's greedy approach, and which are they?

hard

A. 4 activities: {0,3}, {3,5}, {5,9}, {8,10}

B. 4 activities: {0,3}, {3,5}, {4,7}, {8,10}

C. 5 activities: {0,3}, {1,4}, {3,5}, {4,7}, {8,10}

D. 3 activities: {0,3}, {3,5}, {5,9}

Solution

1. Step 1: Sort activities by finish time

   Sorted: [{0,3}, {1,4}, {3,5}, {4,7}, {5,9}, {8,10}]

2. Step 2: Select greedily

   {0,3} (last=3); skip {1,4}; {3,5} (last=5); {4,7} overlaps with {3,5} so skip; {5,9} (last=9); {8,10} overlaps with {5,9} so skip. Actually, {8,10} starts before 9, so overlaps. So selected are {0,3}, {3,5}, {5,9} only 3 activities.

3. Final Answer:

   3 activities: {0,3}, {3,5}, {5,9} -> Option D

4. Quick Check:

   Greedy picks non-overlapping max activities [OK]

Hint: Sort by finish time, pick next starting after last finish [OK]

Common Mistakes:

• Selecting overlapping activities
• Miscounting due to off-by-one errors
• Ignoring sorting step