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Problem solution · Python

Total Waviness of Numbers in Range II

Total Waviness of Numbers in Range II: a Python solution using dynamic programming. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Dynamic programming
Source
Kamyu LeetCode Solutions
Length
184 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

Dynamic programming

For Total Waviness of Numbers in Range II, the implementation records answers for smaller states and reuses them to build the requested result without repeating work.

  1. Define precisely what one DP state represents.
  2. Establish the base cases before transitions are evaluated.
  3. Process states in dependency order and combine only already-known values.

Code notes

  • 184 lines of Python from the credited upstream file total-waviness-of-numbers-in-range-ii.py.
  • The implementation visibly relies on sequence storage, hash lookup, cached states.
  • No explicit loop blocks detected.

Complexity

Multiply the number of reachable states by the work performed for each transition, then include the stored state table in memory usage.

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 codeTotal Waviness of Numbers in Range II · PythonPython
Use this to learn the idea, then write your own version.
# Time:  O(logn * 11 * 11 * 2 * 2 * 10)# Space: O(logn * 11 * 11 * 2 * 2) # memoization by dictclass Solution(object):    def totalWaviness(self, num1, num2):        """        :type num1: int        :type num2: int        :rtype: int        """        def count(x):            def dp(i, prev, prev2, zero, tight):                if i == len(s):                    return 1, 0                key = (i, prev, prev2, zero, tight)                if key not in lookup:                    cnt = w = 0                    mx = int(s[i]) if tight else 9                    for d in xrange(mx+1):                        new_tight = tight and (d == int(s[i]))                        new_zero = zero and (d == 0)                        new_prev2 = prev                        new_prev = d if not new_zero else -1                        new_cnt, nw = dp(i+1, new_prev, new_prev2, new_zero, new_tight)                        cnt += new_cnt                        if not zero and prev2 != -1 and (prev2 < prev and prev > d or prev2 > prev and prev < d):                            w += new_cnt                        w += nw                    lookup[key] = (cnt, w)                return lookup[key]             s = str(x)            lookup = {}            return dp(0, -1, -1, True, True)[1]         return count(num2)-count(num1-1)  # Time:  O(logn * 11 * 11 * 2 * 2 * 10)# Space: O(logn * 11 * 11 * 2 * 2)# memoization by listclass Solution2(object):    def totalWaviness(self, num1, num2):        """        :type num1: int        :type num2: int        :rtype: int        """        def count(x):            def encode(i, prev, prev2, zero, tight):                key = i                key = key*(10+1)+(prev+1)                key = key*(10+1)+(prev2+1)                key = key*2+(1 if zero else 0)                key = key*2+(1 if tight else 0)                return key             def dp(i, prev, prev2, zero, tight):                if i == len(s):                    return 1, 0                key = encode(i, prev, prev2, zero, tight)                if lookup[key] is None:                    cnt = w = 0                    mx = int(s[i]) if tight else 9                    for d in xrange(mx+1):                        new_tight = tight and (d == int(s[i]))                        new_zero = zero and (d == 0)                        new_prev2 = prev                        new_prev = d if not new_zero else -1                        new_cnt, nw = dp(i+1, new_prev, new_prev2, new_zero, new_tight)                        cnt += new_cnt                        if not zero and prev2 != -1:                            if (prev2 < prev and prev > d) or (prev2 > prev and prev < d):                                w += new_cnt                        w += nw                    lookup[key] = (cnt, w)                return lookup[key]             s = str(x)            lookup = [None]*(len(s)*(10+1)*(10+1)*2*2)            return dp(0, -1, -1, True, True)[1]         return count(num2)-count(num1-1)  # Time:  O(logn * 11 * 11 * 2 * 2 * 10)# Space: O(11 * 11 * 2 * 2)# dp by dictclass Solution3(object):    def totalWaviness(self, num1, num2):        """        :type num1: int        :type num2: int        :rtype: int        """        def count(x):            s = str(x)            dp = {}            for prev in xrange(-1, 10):                for prev2 in xrange(-1, 10):                    for zero in xrange(2):                        for tight in xrange(2):                            dp[(prev, prev2, zero, tight)] = (1, 0)            for i in reversed(xrange(len(s))):                new_dp = {}                for prev in xrange(-1, 10):                    for prev2 in xrange(-1, 10):                        for zero in xrange(2):                            for tight in xrange(2):                                cnt = w = 0                                mx = int(s[i]) if tight else 9                                for d in xrange(mx+1):                                    new_tight = tight and (d == int(s[i]))                                    new_zero = zero and (d == 0)                                    new_prev2 = prev                                    new_prev = d if not new_zero else -1                                    key = (new_prev, new_prev2, new_zero, new_tight)                                    if key in dp:                                        new_cnt, nw = dp[key]                                        cnt += new_cnt                                        if not zero and prev2 != -1 and ((prev2 < prev and prev > d) or (prev2 > prev and prev < d)):                                            w += new_cnt                                        w += nw                                new_dp[(prev, prev2, zero, tight)] = (cnt, w)                dp = new_dp            return dp[(-1, -1, True, True)][1]         return count(num2)-count(num1-1)  # Time:  O(logn * 11 * 11 * 2 * 2 * 10)# Space: O(11 * 11 * 2 * 2)# dp by listclass Solution4(object):    def totalWaviness(self, num1, num2):        """        :type num1: int        :type num2: int        :rtype: int        """        def count(x):            def encode(prev, prev2, zero, tight):                key = prev+1                key = key*(10+1)+(prev2+1)                key = key*2+(1 if zero else 0)                key = key*2+(1 if tight else 0)                return key             s = str(x)            state_size = (10+1)*(10+1)*2*2            dp = [None]*state_size            for prev in xrange(-1, 10):                for prev2 in xrange(-1, 10):                    for zero in xrange(2):                        for tight in xrange(2):                            key = encode(prev, prev2, zero, tight)                            dp[key] = (1, 0)            for i in reversed(xrange(len(s))):                new_dp = [None]*state_size                for prev in xrange(-1, 10):                    for prev2 in xrange(-1, 10):                        for zero in xrange(2):                            for tight in xrange(2):                                cnt = w = 0                                mx = int(s[i]) if tight else 9                                for d in xrange(mx+1):                                    new_tight = tight and (d == int(s[i]))                                    new_zero = zero and (d == 0)                                    new_prev2 = prev                                    new_prev = d if not new_zero else -1                                    key = encode(new_prev, new_prev2, new_zero, new_tight)                                    if dp[key] is not None:                                        new_cnt, nw = dp[key]                                        cnt += new_cnt                                        if not zero and prev2 != -1 and ((prev2 < prev and prev > d) or (prev2 > prev and prev < d)):                                            w += new_cnt                                        w += nw                                new_dp[encode(prev, prev2, zero, tight)] = (cnt, w)                dp, new_dp = new_dp, dp            return dp[encode(-1, -1, True, True)][1]         return count(num2)-count(num1-1)

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