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Problem solution · Python

Count the Number of Ideal Arrays

Count the Number of Ideal Arrays: a Python solution using hash-based lookup. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Hash-based lookup
Source
Kamyu LeetCode Solutions
Length
93 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

Hash-based lookup

For Count the Number of Ideal Arrays, the implementation stores previously seen values or frequencies in a hash table for direct membership and lookup operations.

  1. Decide the key that represents the information needed later.
  2. Update its count or stored state while scanning the input.
  3. Use constant-time expected lookups to detect matches or assemble the result.

Code notes

  • 93 lines of Python from the credited upstream file count-the-number-of-ideal-arrays.py.
  • The implementation visibly relies on sequence storage, hash lookup.
  • No explicit loop blocks detected.

Complexity

Expected hash operations are constant time, but the surrounding scan and the number of stored keys determine total work and memory.

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 codeCount the Number of Ideal Arrays · PythonPython
Use this to learn the idea, then write your own version.
# Time:  O(sqrt(m) + n + m * (logm + pi(sqrt(m)))) = O(sqrt(m) + n + m * (logm + sqrt(m)/log(sqrt(m)))), pi(n) = number of primes in a range [1, n] = O(n/logn) by prime number theorem, see https://en.wikipedia.org/wiki/Prime_number_theorem# Space: O(sqrt(m) + n + logm) import collections  # dp, factorization, combinatoricsclass Solution(object):    def idealArrays(self, n, maxValue):        """        :type n: int        :type maxValue: int        :rtype: int        """        MOD = 10**9+7        fact, inv, inv_fact = [[1]*2 for _ in xrange(3)]        def nCr(n, k):            while len(inv) <= n:  # lazy initialization                fact.append(fact[-1]*len(inv) % MOD)                inv.append(inv[MOD%len(inv)]*(MOD-MOD//len(inv)) % MOD)  # https://cp-algorithms.com/algebra/module-inverse.html                inv_fact.append(inv_fact[-1]*inv[-1] % MOD)            return (fact[n]*inv_fact[n-k] % MOD) * inv_fact[k] % MOD         def linear_sieve_of_eratosthenes(n):  # Time: O(n), Space: O(n)            primes = []            spf = [-1]*(n+1)  # the smallest prime factor            for i in xrange(2, n+1):                if spf[i] == -1:                    spf[i] = i                    primes.append(i)                for p in primes:                    if i*p > n or p > spf[i]:                        break                    spf[i*p] = p            return primes         def prime_factors(x):            factors = collections.Counter()            for p in primes:                if p*p > x:                    break                while x%p == 0:                    factors[p] += 1                    x //= p            if x != 1:                factors[x] += 1            return factors         primes = linear_sieve_of_eratosthenes(int(maxValue**0.5))        result = 0        for k in xrange(1, maxValue+1):            total = 1            for c in prime_factors(k).itervalues():                total = (total*nCr(n+c-1, c))%MOD  # H(n, c) = nCr(n+c-1, n)            result = (result+total)%MOD        return result  # Time:  O(n * mlogm)# Space: O(n + m)import collections  # dp, combinatoricsclass Solution2(object):    def idealArrays(self, n, maxValue):        """        :type n: int        :type maxValue: int        :rtype: int        """        MOD = 10**9+7        fact, inv, inv_fact = [[1]*2 for _ in xrange(3)]        def nCr(n, k):            while len(inv) <= n:  # lazy initialization                fact.append(fact[-1]*len(inv) % MOD)                inv.append(inv[MOD%len(inv)]*(MOD-MOD//len(inv)) % MOD)  # https://cp-algorithms.com/algebra/module-inverse.html                inv_fact.append(inv_fact[-1]*inv[-1] % MOD)            return (fact[n]*inv_fact[n-k] % MOD) * inv_fact[k] % MOD         result = 0        dp = collections.Counter(xrange(1, maxValue+1))        for i in xrange(n):             new_dp = collections.Counter()            total = 0            for x, c in dp.iteritems():                total = (total+c)%MOD                for y in xrange(x+x, maxValue+1, x):                     new_dp[y] += c            result = (result+total*nCr(n-1, i))%MOD            dp = new_dp        return result

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