Minimum Time to Finish the Race

arraydynamic programming

Problem Statement

You are given a 0-indexed 2D integer array tires where tires[i] = [f _i , r _i ] indicates that the i ^th tire can finish its x ^th successive lap in f _i * r _i ^(x-1) seconds.

For example, if f _i = 3 and r _i = 2 , then the tire would finish its 1 ^st lap in 3 seconds, its 2 ^nd lap in 3 2 = 6 seconds, its 3 ^rd lap in 3 2 ^2 = 12 seconds, etc.

You are also given an integer changeTime and an integer numLaps .

The race consists of numLaps laps and you may start the race with any tire. You have an unlimited supply of each tire and after every lap, you may change to any given tire (including the current tire type) if you wait changeTime seconds.

Return the minimum time to finish the race.

Examples

Example 1

Input: tires = [[2,3],[3,4]], changeTime = 5, numLaps = 4
Output: 21
Lap 1: Start with tire 0 and finish the lap in 2 seconds.
Lap 2: Continue with tire 0 and finish the lap in 2 * 3 = 6 seconds.
Lap 3: Change tires to a new tire 0 for 5 seconds and then finish the lap in another 2 seconds.
Lap 4: Continue with tire 0 and finish the lap in 2 * 3 = 6 seconds.
Total time = 2 + 6 + 5 + 2 + 6 = 21 seconds.
The minimum time to complete the race is 21 seconds.

Example 2

Input: tires = [[1,10],[2,2],[3,4]], changeTime = 6, numLaps = 5
Output: 25
Lap 1: Start with tire 1 and finish the lap in 2 seconds.
Lap 2: Continue with tire 1 and finish the lap in 2 * 2 = 4 seconds.
Lap 3: Change tires to a new tire 1 for 6 seconds and then finish the lap in another 2 seconds.
Lap 4: Continue with tire 1 and finish the lap in 2 * 2 = 4 seconds.
Lap 5: Change tires to tire 0 for 6 seconds then finish the lap in another 1 second.
Total time = 2 + 4 + 6 + 2 + 4 + 6 + 1 = 25 seconds.
The minimum time to complete the race is 25 seconds.

Constraints

1 <= tires.length <= 10 ^5

tires[i].length == 2

1 <= f _i , changeTime <= 10 ^5

2 <= r _i <= 10 ^5

1 <= numLaps <= 1000

😤

Hard

Difficulty

Topic Info

ModuleDSA

CategoryArray

Sub-topicDynamic Programming