Optimize a Python Function for Speed

Write a Python function that takes a large array of integers as input. This function should iterate over the array, find pairs of numbers that add up to a target value, and store them in a dictionary where the keys are the pairs and the values are their frequencies. The target value is the sum of all the elements in the array divided by 2. The function should return the dictionary. Use a single loop to iterate over the array, and employ techniques such as memoization or caching to improve performance. The input array will contain at least 100,000 elements and will be randomly generated.