Distances

Your task is to implement a class that maintains a list of numbers and has a method for efficiently computing the sum of distances between all pairs of occurrences of a given number.

For example, when the list is $[1,2,1,3,3,1,2,1]$, the distances for the number $1$ are:

• positions $0$ and $2$: distance $2$
• positions $0$ and $5$: distance $5$
• positions $0$ and $7$: distance $7$
• positions $2$ and $5$: distance $3$
• positions $2$ and $7$: distance $5$
• positions $5$ and $7$: distance $2$

Thus the sum of the distances is $2+5+7+3+5+2=24$.

In a file distances.py, implement the class DistanceTracker with the methods:

• append(number): add number to the end of the list
• sum(number): return the sum of distances for number

Both methods should work in $O(1)$ time.

class DistanceTracker:
    def __init__(self):
        # TODO

    def append(self, number):
        # TODO

    def sum(self, number):
        # TODO

if __name__ == "__main__":
    tracker = DistanceTracker()

    tracker.append(1)
    tracker.append(2)
    tracker.append(1)
    tracker.append(3)
    tracker.append(3)
    tracker.append(1)
    tracker.append(2)
    tracker.append(1)

    print(tracker.sum(1)) # 24
    print(tracker.sum(2)) # 5
    print(tracker.sum(3)) # 1

    tracker.append(1)
    tracker.append(2)
    tracker.append(3)

    print(tracker.sum(1)) # 42
    print(tracker.sum(2)) # 16
    print(tracker.sum(3)) # 14

If a number does not appear on the list or appears only once, the desired result is $0$:

    tracker = DistanceTracker()
    print(tracker.sum(1)) # 0
    tracker.append(1)
    print(tracker.sum(1)) # 0

You can test the efficiency of your solution with the following code. In this case too the code should finish almost immediately.

    tracker = DistanceTracker()
    total = 0
    for i in range(10**5):
        tracker.append(1)
        total += tracker.sum(1)
    print(total) # 4166749999583325000