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Problem solution · Python

Sum of Beautiful Subsequences

Sum of Beautiful Subsequences: a Python solution using segment tree or range structure. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Segment tree or range structure
Source
Kamyu LeetCode Solutions
Length
100 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Segment tree or range structure

For Sum of Beautiful Subsequences, the implementation stores interval information in a range-query data structure so updates and queries avoid rescanning the full input.

  1. Choose the aggregate stored for each interval or prefix.
  2. Build or initialize the structure from the input.
  3. Apply updates and combine the affected nodes to answer each query.

Code notes

  • 100 lines of Python from the credited upstream file sum-of-beautiful-subsequences.py.
  • The implementation visibly relies on sequence storage, ordered lookup.
  • No explicit loop blocks detected.

Complexity

Count the build once, then multiply the logarithmic update or query path by the number of operations.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from Kamyu LeetCode Solutions by kamyu104 and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeSum of Beautiful Subsequences · PythonPython
Use this to learn the idea, then write your own version.
# Time:  precompute: O(rlogr), r = max_nums#        runtime:    O(mx + nlogr * (log(nlogr) + logn)), mx = max(nums)# Space: O(rlogr) # number theory, bit, fenwick treeMOD = 10**9+7class BIT(object):  # 0-indexed.    def __init__(self, n):        self.__bit = [0]*(n+1)  # Extra one for dummy node.     def add(self, i, val):        i += 1  # Extra one for dummy node.        while i < len(self.__bit):            self.__bit[i] = (self.__bit[i]+val) % MOD            i += (i & -i)     def query(self, i):        i += 1  # Extra one for dummy node.        ret = 0        while i > 0:            ret = (ret+self.__bit[i]) % MOD            i -= (i & -i)        return ret  def factors(n):  # Time: O(nlogn)    result = [[] for _ in xrange(n+1)]    for i in xrange(1, n+1):        for j in range(i, n+1, i):            result[j].append(i)    return result  def phi_sieve(n):  # Time: O(nlog(logn))    phi = range(n+1)    for i in xrange(2, n+1):        if phi[i] != i:            continue        for j in xrange(i, n+1, i):            phi[j] -= phi[j]//i    return phi  MAX_NUM = 7 * 10**4FACTORS = factors(MAX_NUM)PHI = phi_sieve(MAX_NUM)class Solution(object):    def totalBeauty(self, nums):        """        :type nums: List[int]        :rtype: int        """        def count(arr):            for i, x in enumerate(sorted(arr)):  # coordinate compression                val_to_idx[x] = i            bit = BIT(len(arr))            for x in arr:                bit.add(val_to_idx[x], bit.query(val_to_idx[x]-1)+1)            return bit.query(len(arr)-1)         mx = max(nums)        val_to_idx = [0]*(mx+1)        lookup = [[] for _ in xrange(mx+1)]        for x in nums:            for d in FACTORS[x]:                lookup[d].append(x)        return reduce(lambda accu, x: (accu+x)%MOD, (PHI[g]*count(lookup[g]) for g in reversed(xrange(1, mx+1))), 0)  # Time:  precompute: O(rlogr), r = max_nums#        runtime:    O(mx * log(mx) + nlogr * (log(nlogr) + logn)), mx = max(nums)# Space: O(rlogr)# number theory, bit, fenwick treeclass Solution2(object):    def totalBeauty(self, nums):        """        :type nums: List[int]        :rtype: int        """        def count(arr):            val_to_idx = {x:i for i, x in enumerate(sorted(set(arr)))}  # coordinate compression            bit = BIT(len(val_to_idx))            for x in arr:                bit.add(val_to_idx[x], bit.query(val_to_idx[x]-1)+1)            return bit.query(len(val_to_idx)-1)         mx = max(nums)        lookup = [[] for _ in xrange(mx+1)]        for x in nums:            for d in FACTORS[x]:                lookup[d].append(x)        result = 0        cnt = [0]*(mx+1)        for g in reversed(xrange(1, mx+1)):            cnt[g] = count(lookup[g])            for ng in xrange(g+g, mx+1, g):                cnt[g] -= cnt[ng]            result = (result+g*cnt[g])%MOD        return result

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