
    Design an algorithm to determine the optimal bus route segment that maximizes total passenger pickup while ensuring stops are arranged in increasing passenger density order, from the start of the route up to a specific current stop, and must include a designated high-demand future stop. The route must maintain a strictly increasing pattern of passenger counts between consecutive selected stops to prevent overcrowding. Given an array of passenger counts at each stop, the current stop position, and a mandatory future stop position, find the route segment that yields the maximum total passenger count while satisfying these constraints. The solution should help transit authorities optimize route planning while ensuring service to key high-volume locations.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm to determine the optimal bus route segment that maximizes total passenger pickup while ensuring stops are arranged in increasing passenger density order, from the start of the route up to a specific current stop, and must include a designated high-demand future stop. The route must maintain a strictly increasing pattern of passenger counts between consecutive selected stops to prevent overcrowding. Given an array of passenger counts at each stop, the current stop position, and a mandatory future stop position, find the route segment that yields the maximum total passenger count while satisfying these constraints. The solution should help transit authorities optimize route planning while ensuring service to key high-volume locations.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm to determine the optimal bus route segment that maximizes total passenger pickup while ensuring stops are arranged in increasing passenger density order, from the start of the route up to a specific current stop, and must include a designated high-demand future stop. The route must maintain a strictly increasing pattern of passenger counts between consecutive selected stops to prevent overcrowding. Given an array of passenger counts at each stop, the current stop position, and a mandatory future stop position, find the route segment that yields the maximum total passenger count while satisfying these constraints. The solution should help transit authorities optimize route planning while ensuring service to key high-volume locations.
    

    """
    dp = [[0 for i in range(n)] for i in range(n)]
    for i in range(n):
        if a[i] > a[0]:
            dp[0][i] = a[i] + a[0]
        else:
            dp[0][i] = a[i]
    for i in range(1, n):
        for j in range(n):
            if a[j] > a[i] and j > i:
                if dp[i - 1][i] + a[j] > dp[i - 1][j]:
                    dp[i][j] = dp[i - 1][i] + a[j]
                else:
                    dp[i][j] = dp[i - 1][j]
            else:
                dp[i][j] = dp[i - 1][j]
    return dp[index][k]