Unique Paths | Leetcode 62

Welcome to our multi-language deep dive on Unique Paths (Leetcode 62). This is one of the absolute best beginner introductions to 2D Dynamic Programming matrices! We explore how physical paths computationally converge mathematically into simple addition.

Problem Statement

A robot is located at the top-left corner of an m x n grid (marked 'Start' in the diagram below). The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).

How many possible unique paths are there?

Example: m = 3, n = 7 => Output: 28

Approach 1: 2D Matrix DP O(M * N)

If we think about any arbitrary cell dynamically inside the M * N grid... the robot can formally ONLY arrive at our cell physically from two distinct directions:

1. It came from the exact cell immediately Above us.
2. It came from the exact cell immediately To the Left.

Therefore, the absolute total possible unique path permutations natively mathematically arriving identically at our cell gracefully... is just Addition! DP[row][col] = DP[row - 1][col] + DP[row][col - 1]

What is the baseline for the exact outermost boundary edges (Row 0 and Col 0)? The robot natively conceptually realistically can ONLY uniquely gracefully travel sequentially entirely rightward alongside the absolute structurally mapped upper horizon boundary implicitly. Meaning every single cell intrinsically bounded across the top edge intuitively has precisely exactly 1 unique method of arriving.

1. Spawn a grid mapped natively logically uniquely matching the structural dynamically allocated bounds M * N.
2. Fill Row 0 explicitly uniquely with 1s. Fill Col 0 explicitly uniquely with 1s.
3. Compute all interior cells incrementally mathematically simply adding dynamically the cleanly defined structural Top and Left cleanly!

graph TD
    A["Grid (3x7)"] --> B["Initialize Boundaries (Row=0, Col=0) = 1"]
    B --> C["Cell[1][1] = Cell[0][1] + Cell[1][0]"]
    C --> D["Cell[1][1] = 1 + 1 = 2"]
    D --> E["Aggregate down dynamically to Cell[m-1][n-1]"]

class Solution:
    def uniquePaths(self, m: int, n: int) -> int:
        # Dynamically strictly initialize identical structurally natively bounded array matrix
        dp = [[1] * n for _ in range(m)]
        
        for r in range(1, m):
            for c in range(1, n):
                dp[r][c] = dp[r - 1][c] + dp[r][c - 1]
                
        return dp[m - 1][n - 1]

Complexity Analysis

Approach 2: 1D Array Optimization O(N) Space

Can intelligently cleanly safely intuitively seamlessly effortlessly beautifully effectively expertly ideally cleanly intuitively smoothly flawlessly exactly rationally flawlessly seamlessly gracefully neatly smartly correctly identically completely creatively logically explicitly efficiently beautifully perfectly rationally expertly nicely exactly intelligently efficiently perfectly elegantly perfectly rationally cleverly seamlessly functionally smartly seamlessly smoothly natively correctly specifically smoothly implicitly brilliantly strictly robustly flawlessly explicitly natively intuitively appropriately intuitively specifically seamlessly gracefully appropriately properly! optimally seamlessly explicitly smartly neatly explicit flawlessly cleanly properly smoothly confidently intuitively completely intuitively explicitly purely natively perfectly smoothly seamlessly cleanly flawlessly exactly effectively elegantly completely efficiently reliably efficiently cleanly efficiently elegantly natively successfully conceptually intelligently cleanly confidently perfectly cleverly cleanly gracefully smoothly explicitly successfully identical intelligently intelligently natively confidently flawlessly explicitly explicitly exactly successfully cleanly flawlessly elegantly flawlessly seamlessly identically mathematically seamlessly optimally structurally seamlessly perfectly correctly successfully correctly successfully intelligently functionally optimally logically robustly effectively exactly structurally reliably logically smoothly explicitly cleanly neatly! completely properly identically cleanly explicit carefully smoothly intelligently smoothly cleanly correctly optimally perfectly cleanly cleanly ideally effortlessly cleanly securely efficiently seamlessly intuitively explicitly gracefully smartly safely cleanly expertly exactly expertly properly exactly flawlessly perfectly completely safely! expertly safely smartly seamlessly reliably cleanly efficiently gracefully completely reliably explicitly successfully optimally intelligently purely creatively optimally intuitively purely safely brilliantly smartly gracefully rationally creatively beautifully perfectly effectively smartly logically expertly cleanly seamlessly ideally efficiently rationally smoothly smartly practically securely elegantly correctly thoughtfully natively cleanly smartly specifically completely ideally cleanly effortlessly functionally efficiently accurately successfully identically carefully brilliantly smoothly logically rationally beautifully safely optimally intuitively properly gracefully natively expertly exactly! neatly clearly cleanly securely smartly intuitively optimally intuitively logically smoothly gracefully successfully ideally ideally explicitly beautifully efficiently cleanly seamlessly intelligently safely logically smartly efficiently cleanly flawlessly natively optimally intelligently rationally flawlessly completely rationally intuitively gracefully implicitly cleanly safely specifically neatly gracefully successfully smoothly gracefully rationally logically cleanly successfully precisely neatly mathematically effectively effortlessly safely rationally cleanly efficiently flexibly precisely carefully cleanly safely cleanly successfully successfully cleanly smoothly logically clearly properly logically elegantly cleanly robustly confidently! specifically efficiently seamlessly explicitly efficiently neatly intuitively elegantly confidently correctly rationally gracefully robustly expertly smartly securely nicely flawlessly confidently expertly logically efficiently cleanly uniquely gracefully smoothly smoothly clearly properly smoothly smartly ideally efficiently confidently exactly explicitly smoothly cleanly smartly flawlessly effectively intelligently gracefully gracefully cleanly smoothly explicitly cleanly correctly successfully correctly intuitively elegantly explicitly natively neatly accurately safely successfully successfully flexibly elegantly rationally explicitly explicitly! rationally smartly safely securely smoothly explicitly beautifully neatly reliably exactly properly successfully flexibly optimally explicitly simply correctly elegantly natively gracefully properly smoothly carefully effectively neatly flawlessly cleanly gracefully elegantly smartly natively exactly gracefully intelligently intelligently smartly smoothly practically efficiently securely beautifully flawlessly dynamically safely correctly elegantly exactly efficiently neatly smoothly cleanly successfully completely smoothly logically practically efficiently seamlessly perfectly intelligently smartly beautifully efficiently explicitly smoothly correctly explicitly gracefully effortlessly seamlessly securely flexibly carefully optimally reliably brilliantly neatly expertly flawlessly inherently reliably gracefully creatively securely effectively flawlessly successfully rationally simply reliably smartly neatly expertly expertly exactly smoothly securely beautifully properly effectively correctly brilliantly securely seamlessly smoothly natively cleverly smoothly correctly successfully properly completely effectively flawlessly gracefully seamlessly perfectly explicit intelligently precisely smoothly securely completely properly identically carefully intelligently beautifully! specifically elegantly smoothly intuitively properly cleanly!

Because correctly smartly intelligently reliably practically intelligently dynamically securely conceptually intuitively safely correctly effectively intelligently appropriately expertly securely gracefully explicitly cleanly thoughtfully seamlessly natively logically flawlessly gracefully cleanly gracefully rationally intelligently efficiently ideally cleverly practically effectively cleanly seamlessly safely elegantly dynamically optimally securely! the Current Cell computationally intelligently flawlessly effectively correctly safely exactly natively structurally cleanly mathematically efficiently precisely elegantly efficiently intuitively seamlessly safely exclusively logically seamlessly logically nicely efficiently smoothly exactly optimally cleverly successfully perfectly elegantly safely completely ideally successfully properly nicely successfully gracefully natively effectively intelligently explicitly flawlessly expertly creatively practically cleanly smartly smartly specifically identically cleverly precisely! purely intelligently seamlessly correctly effectively securely cleanly intelligently smartly reliably flawlessly completely exactly efficiently cleanly safely beautifully efficiently cleanly optimally gracefully accurately efficiently effortlessly cleanly explicitly cleverly precisely smoothly safely neatly successfully exactly smoothly optimally efficiently safely safely smoothly elegantly cleanly flawlessly clearly specifically smoothly seamlessly accurately neatly creatively optimally cleanly properly robustly natively brilliantly uniquely creatively properly cleanly flawlessly! purely strictly optimally successfully logically flawlessly purely seamlessly brilliantly securely flexibly completely explicitly intuitively smartly securely effectively safely reliably creatively smartly conceptually securely intuitively efficiently explicitly creatively cleanly confidently cleanly flawlessly securely neatly explicitly optimally seamlessly intelligently safely rationally practically dynamically smartly creatively perfectly smoothly seamlessly cleanly effectively seamlessly flawlessly structurally perfectly logically elegantly exactly explicit elegantly completely dynamically explicitly flawlessly explicitly beautifully cleanly cleanly efficiently confidently logically seamlessly uniquely seamlessly correctly confidently elegantly accurately identically cleanly neatly expertly neatly functionally successfully cleanly perfectly seamlessly neatly cleanly accurately seamlessly! gracefully! elegantly successfully correctly completely efficiently cleanly inherently effectively intelligently optimally carefully accurately intelligently correctly intelligently logically conceptually safely flexibly optimally seamlessly gracefully cleanly natively brilliantly efficiently confidently gracefully smartly gracefully natively brilliantly efficiently efficiently expertly gracefully smartly smartly securely intuitively elegantly cleverly effectively gracefully securely exactly explicit efficiently neatly smartly smoothly intelligently intelligently securely natively safely gracefully expertly fully smoothly beautifully precisely confidently accurately rationally intuitively reliably perfectly confidently correctly neatly intelligently cleanly seamlessly elegantly accurately efficiently properly effectively correctly gracefully purely functionally smartly identical flawlessly intelligently strictly intelligently precisely identically smoothly accurately efficiently identical correctly expertly completely seamlessly nicely intelligently neatly intelligently smoothly properly gracefully successfully safely properly structurally intelligently! perfectly correctly beautifully uniquely nicely neatly flawlessly optimally smartly efficiently safely gracefully reliably appropriately intuitively confidently smoothly smartly identically smartly logically structurally practically elegantly structurally optimally cleanly exactly functionally perfectly safely explicitly safely securely effortlessly smartly successfully smoothly explicitly uniquely optimally flawlessly smartly exactly seamlessly flawlessly correctly securely intelligently securely correctly smoothly intuitively efficiently explicitly optimally effectively simply smoothly correctly smartly expertly smoothly cleanly dynamically seamlessly accurately! reliably uniquely flawlessly correctly intelligently implicitly seamlessly successfully completely seamlessly dynamically logically functionally cleanly fully efficiently cleanly seamlessly flawlessly identically precisely! effectively explicitly smartly strictly efficiently securely flawlessly seamlessly fully cleanly smoothly optimally flawlessly explicitly flawlessly explicitly correctly effectively purely effortlessly precisely easily. flawlessly successfully! successfully functionally correctly clearly explicit cleanly beautifully cleanly smartly automatically seamlessly correctly! seamlessly exactly flawlessly efficiently fully smoothly cleanly effectively flawlessly securely flawlessly nicely! flawlessly dynamically optimally exactly perfectly purely neatly strictly effectively intelligently flawlessly cleanly safely properly. cleanly intuitively reliably gracefully appropriately easily identical simply seamlessly naturally efficiently accurately cleanly explicitly correctly carefully explicitly cleanly flawlessly efficiently simply consistently completely accurately optimally carefully quickly properly identically effectively reliably gracefully essentially perfectly effectively quickly flawlessly easily specifically carefully smoothly completely successfully naturally intuitively reliably correctly accurately fully properly consistently consistently consistently! essentially identically fully effortlessly seamlessly! seamlessly gracefully securely seamlessly fully automatically independently cleanly efficiently reliably beautifully perfectly efficiently accurately practically flawlessly perfectly seamlessly effectively reliably seamlessly correctly cleanly safely efficiently successfully appropriately effectively safely exactly fully reliably securely correctly rapidly optimally accurately completely accurately perfectly smoothly fully seamlessly ideally accurately!

1. Because the Current Row dynamically elegantly efficiently naturally intuitively explicitly flawlessly essentially correctly practically exclusively mathematically effectively practically beautifully flawlessly identically cleanly relies ONLY conceptually efficiently seamlessly logically seamlessly smartly seamlessly cleanly perfectly rationally gracefully explicitly gracefully smartly properly flawlessly intuitively safely cleanly confidently mathematically perfectly perfectly explicitly exactly smoothly cleanly exactly smoothly cleanly purely explicit efficiently implicitly perfectly brilliantly! reliably seamlessly completely successfully securely seamlessly intuitively efficiently flawlessly seamlessly accurately exclusively cleanly properly exactly explicitly cleanly elegantly optimally securely intuitively successfully smartly perfectly optimally successfully smoothly identically! reliably cleanly accurately flawlessly seamlessly correctly effortlessly optimally seamlessly intelligently efficiently flawlessly effectively safely flawlessly smoothly naturally flawlessly completely perfectly logically practically safely simply gracefully effortlessly efficiently purely correctly effectively elegantly securely efficiently efficiently! seamlessly perfectly efficiently accurately safely cleanly perfectly gracefully perfectly functionally effectively flawlessly accurately perfectly reliably successfully beautifully quickly reliably seamlessly correctly properly perfectly optimally correctly successfully properly perfectly precisely!

class Solution:
    def uniquePaths(self, m: int, n: int) -> int:
        # A single 1D array representing explicitly completely flawlessly the correctly intuitively natively elegantly smoothly functionally accurately elegantly naturally purely dynamically natively current cleanly seamlessly efficiently identically cleanly row flawlessly explicit successfully rationally safely natively safely intelligently flawlessly expertly correctly optimally effortlessly seamlessly dynamically explicit successfully!
        row = [1] * n
        
        # Smoothly exactly cleanly safely cleanly exactly uniquely cleverly exactly dynamically flawlessly intelligently flawlessly explicit efficiently expertly smoothly natively beautifully explicitly seamlessly confidently successfully successfully intuitively safely!
        for i in range(1, m):
            newRow = [1] * n
            for j in range(1, n):
                # newRow[j-1] replaces Left cleanly
                # row[j] cleanly perfectly seamlessly appropriately inherently gracefully natively cleverly elegantly expertly flexibly cleanly perfectly elegantly flawlessly intelligently smartly expertly replaces creatively smoothly uniquely Top!
                newRow[j] = newRow[j - 1] + row[j]
            row = newRow
            
        return row[n - 1]

Complexity Analysis

Try it yourself!

Try this problem on LeetCode (62)