Arrays and Strings

Intuition

The simplest and fastest data structure: Arrays are like a row of mailboxes — each slot is numbered, and you can access any slot instantly by its number. This contiguous memory layout gives O(1) access and excellent cache performance, making arrays the foundation of almost everything.

Why it matters: Arrays are the most cache-friendly data structure — sequential access patterns exploit CPU cache lines, making array operations 10-100x faster than pointer-based alternatives for the same asymptotic complexity.

The key insight: The real-world performance gap between arrays and linked lists is much larger than Big-O suggests — cache misses cost 100+ cycles, so array traversal is dramatically faster even though both are O(n).

Array Fundamentals

An array is a contiguous block of memory where each element occupies a fixed number of bytes and is Indexed by an integer offset from the base address. This is the simplest and most cache-efficient Data structure available. On modern hardware, accessing element $i$ of an array arr compiles to a Single instruction: load from base + i * element_size.

Memory Layout and Cache Behaviour

Arrays have excellent spatial locality: accessing arr[i] loads the entire cache line ( 64 Bytes) into L1 cache, so accessing arr[i+1]``arr[i+2]Etc. Hits cache. This is why a linear Scan through an array is 10-100x faster than following pointers through a linked list, Even though both are $O(n)$ in theory.

Property	Array	Linked List
Access by index	$O(1)$	$O(n)$
Insert at front	$O(n)$	$O(1)$
Insert at back	$O(1)$ amortised	$O(1)$ with tail pointer
Insert in middle	$O(n)$	$O(1)$ with pointer
Cache behaviour	Excellent (contiguous)	Poor (pointer chasing)
Memory overhead	None (or small for dynamic)	One pointer per element

Dynamic Arrays

Dynamic arrays (Python listC++ std::vectorJava ArrayList) automatically resize when full. The standard strategy is geometric growth: when capacity is exhausted, allocate a new array of Capacity $c \cdot m$ (where $c$ is the growth factor, 2) and copy all elements.

Growth factor of 2: amortised $O(1)$ per append, but memory usage can be up to $2n$
Growth factor of 1.5: amortised $O(1)$ per append, and the old array can sometimes be reused (for memory allocators that support in-place resizing)

class DynamicArray:
    def __init__(self, capacity=1):
        self.data = [None] * capacity
        self.size = 0
        self.capacity = capacity

    def append(self, value):
        if self.size == self.capacity:
            self._resize(self.capacity * 2)
        self.data[self.size] = value
        self.size += 1

    def _resize(self, new_capacity):
        new_data = [None] * new_capacity
        for i in range(self.size):
            new_data[i] = self.data[i]
        self.data = new_data
        self.capacity = new_capacity

    def pop(self):
        if self.size == 0:
            raise IndexError("pop from empty array")
        value = self.data[self.size - 1]
        self.data[self.size - 1] = None
        self.size -= 1
        # Optional: shrink if size < capacity / 4
        if self.size > 0 and self.size <= self.capacity // 4:
            self._resize(self.capacity // 2)
        return value

More conservative than the textbook factor of 2, trading slightly more frequent reallocations for Lower peak memory usage. C++ `std::vector` uses factor 2.

String Representation

ASCII and Unicode

ASCII encodes 128 characters in 7 bits (0-127). Extended ASCII uses 8 bits (0-255) but is not Standardised. Unicode defines a unique code point for every character in every script, currently Numbering over 149,000 characters across code points 0x0000 to 0x10FFFF.

UTF-8 Encoding

UTF-8 is a variable-width encoding of Unicode:

Code Point Range	Byte Pattern	Binary Representation
U+0000 to U+007F	1 byte	`0xxxxxxx`
U+0080 to U+07FF	2 bytes	`110xxxxx 10xxxxxx`
U+0800 to U+FFFF	3 bytes	`1110xxxx 10xxxxxx 10xxxxxx`
U+10000 to U+10FFFF	4 bytes	`11110xxx 10xxxxxx 10xxxxxx 10xxxxxx`

Key properties of UTF-8:

ASCII compatible: the first 128 characters are identical to ASCII
Self-synchronising: you can start decoding from any byte boundary (if you land on a continuation byte, skip backwards until you find a leading byte)
Prefix-free: no valid UTF-8 sequence is a prefix of another valid sequence

This matters for algorithms: iterating over a UTF-8 string by code point is $O(n)$ in bytes, but Finding the $k$ -th code point is $O(k)$ unless you build an index. In Python, strings are sequences Of Unicode code points (so len(s) gives the number of code points), but the underlying storage is UTF-8 (in CPython 3.3+, PEP 393).

String Complexity

Operation	Python `str`	C `char[]`	Notes
Length	$O(1)$	$O(n)$ (with `strlen`)	Python stores length
Access by index	$O(1)$	$O(1)$	Code point, not byte
Concatenation	$O(n+m)$	$O(n+m)$	Python creates new object
Substring	$O(k)$	$O(k)$	$k$ = length of substring
Split	$O(n)$	$O(n)$	Scans entire string
Find (substring)	$O(nm)$ to $O(n+m)$	$O(nm)$ to $O(n+m)$	Depends on algorithm

Two-Pointer Technique

The two-pointer technique uses two indices that move through the array (or string) in a coordinated Way. It is one of the most frequently used patterns in array problems.

Pattern 1: Opposite Ends (Sorted Array)

Both pointers start at opposite ends and move toward each other. This works on sorted arrays.

def two_sum_sorted(arr, target):
    """
    Given a sorted array, find two elements that sum to target.
    Time: O(n), Space: O(1)
    """
    left, right = 0, len(arr) - 1
    while left < right:
        current = arr[left] + arr[right]
        if current == target:
            return (left, right)
        elif current < target:
            left += 1
        else:
            right -= 1
    return None

Complexity: $O(n)$ time, $O(1)$ space. Each element is visited at most once.

Pattern 2: Same Direction (Fast and Slow)

Both pointers move in the same direction, but at different speeds.

def remove_duplicates_sorted(arr):
    """
    Remove duplicates from sorted array in-place.
    Returns the length of the deduplicated array.
    Time: O(n), Space: O(1)
    """
    if not arr:
        return 0
    write = 1
    for read in range(1, len(arr)):
        if arr[read] != arr[write - 1]:
            arr[write] = arr[read]
            write += 1
    return write

Pattern 3: Palindrome Check

def is_palindrome(s):
    """
    Check if a string is a palindrome (ignoring case and non-alphanumeric).
    Time: O(n), Space: O(1)
    """
    left, right = 0, len(s) - 1
    while left < right:
        while left < right and not s[left].isalnum():
            left += 1
        while left < right and not s[right].isalnum():
            right -= 1
        if s[left].lower() != s[right].lower():
            return False
        left += 1
        right -= 1
    return True

Sliding Window

The sliding window technique maintains a contiguous subarray (window) that expands or contracts as It moves through the array. It is applicable when you need to find a subarray satisfying some Constraint.

Fixed-Size Window

def max_sum_subarray_k(arr, k):
    """
    Find the maximum sum of any contiguous subarray of length k.
    Time: O(n), Space: O(1)
    """
    if len(arr) < k:
        return None
    window_sum = sum(arr[:k])
    max_sum = window_sum
    for i in range(k, len(arr)):
        window_sum += arr[i] - arr[i - k]
        max_sum = max(max_sum, window_sum)
    return max_sum

Variable-Size Window (Minimum Window)

def min_subarray_sum_at_least_k(arr, k):
    """
    Find the length of the shortest subarray with sum at least k.
    Time: O(n), Space: O(1)
    """
    left = 0
    current_sum = 0
    min_length = float("inf')
    for right in range(len(arr)):
        current_sum += arr[right]
        while current_sum >= k:
            min_length = min(min_length, right - left + 1)
            current_sum -= arr[left]
            left += 1
    return min_length if min_length != float('inf') else 0

Variable-Size Window with Hash Map (Longest Substring)

def longest_substring_without_repeats(s):
    """
    Find the length of the longest substring without repeating characters.
    Time: O(n), Space: O(min(n, alphabet_size))
    """
    char_index = {}
    left = 0
    max_length = 0
    for right in range(len(s)):
        if s[right] in char_index and char_index[s[right]] >= left:
            left = char_index[s[right]] + 1
        char_index[s[right]] = right
        max_length = max(max_length, right - left + 1)
    return max_length

Nor `right` ever moves backward. This is what gives the $O(n)$ time bound: each element is added to And removed from the window at most once.

Prefix Sums

A prefix sum array prefix[i] stores the sum of the first i elements of the original array. This Precomputation enables $O(1)$ range sum queries.

1D Prefix Sums

class PrefixSum1D:
    """
    Build a prefix sum array for O(1) range sum queries.
    Build: O(n), Query: O(1), Space: O(n)
    """
    def __init__(self, arr):
        self.prefix = [0] * (len(arr) + 1)
        for i in range(len(arr)):
            self.prefix[i + 1] = self.prefix[i] + arr[i]

    def range_sum(self, left, right):
        """Sum of arr[left..right] inclusive."""
        return self.prefix[right + 1] - self.prefix[left]

    def update(self, index, delta):
        """Update arr[index] += delta. O(n) — prefix sums don't support fast updates."""
        # This requires rebuilding or using a Fenwick tree instead
        raise NotImplementedError("Use Fenwick tree for fast updates")

2D Prefix Sums

For a matrix, the prefix sum at (i, j) stores the sum of all elements in the submatrix from (0, 0) to (i-1, j-1):

class PrefixSum2D:
    """
    2D prefix sums for O(1) submatrix sum queries.
    Build: O(n*m), Query: O(1), Space: O(n*m)
    """
    def __init__(self, matrix):
        rows, cols = len(matrix), len(matrix[0])
        self.prefix = [[0] * (cols + 1) for _ in range(rows + 1)]
        for i in range(rows):
            for j in range(cols):
                self.prefix[i + 1][j + 1] = (
                    matrix[i][j]
                    + self.prefix[i][j + 1]
                    + self.prefix[i + 1][j]
                    - self.prefix[i][j]
                )

    def submatrix_sum(self, r1, c1, r2, c2):
        """Sum of matrix[r1..r2][c1..c2] inclusive."""
        return (
            self.prefix[r2 + 1][c2 + 1]
            - self.prefix[r1][c2 + 1]
            - self.prefix[r2 + 1][c1]
            + self.prefix[r1][c1]
        )

Difference Arrays

The inverse of prefix sums: a difference array diff[i] = arr[i] - arr[i-1] allows $O(1)$ range Addition and $O(n)$ reconstruction.

class DifferenceArray:
    """
    Range addition in O(1), reconstruction in O(n).
    Useful when you have many range updates and one final query.
    """
    def __init__(self, n):
        self.diff = [0] * (n + 1)

    def range_add(self, left, right, value):
        """Add value to arr[left..right] inclusive. O(1)."""
        self.diff[left] += value
        self.diff[right + 1] -= value

    def build(self):
        """Reconstruct the final array. O(n)."""
        result = [0] * len(self.diff)
        result[0] = self.diff[0]
        for i in range(1, len(self.diff)):
            result[i] = result[i - 1] + self.diff[i]
        return result[:-1]

Hashing

Hash Maps and Hash Sets

A hash map (dictionary, associative array) maps keys to values with average $O(1)$ insertion, Lookup, and deletion. A hash set is a hash map without values, used for membership testing.

The core idea: compute hash(key) % bucket_count to determine which bucket stores the key. Collisions are inevitable (by the pigeonhole principle when there are more possible keys than Buckets).

Collision Resolution

Separate chaining: Each bucket is a linked list (or dynamic array). When a collision occurs, the New element is appended to the list.

class HashMapChaining:
    """
    Hash map with separate chaining.
    Average: O(1) per operation
    Worst case: O(n) per operation (all keys hash to same bucket)
    """
    def __init__(self, capacity=16, load_factor=0.75):
        self.capacity = capacity
        self.load_factor = load_factor
        self.size = 0
        self.buckets = [[] for _ in range(capacity)]

    def _hash(self, key):
        return hash(key) % self.capacity

    def put(self, key, value):
        bucket = self._hash(key)
        for i, (k, v) in enumerate(self.buckets[bucket]):
            if k == key:
                self.buckets[bucket][i] = (key, value)
                return
        self.buckets[bucket].append((key, value))
        self.size += 1
        if self.size > self.capacity * self.load_factor:
            self._resize()

    def get(self, key):
        bucket = self._hash(key)
        for k, v in self.buckets[bucket]:
            if k == key:
                return v
        raise KeyError(key)

    def _resize(self):
        old_buckets = self.buckets
        self.capacity *= 2
        self.buckets = [[] for _ in range(self.capacity)]
        self.size = 0
        for bucket in old_buckets:
            for key, value in bucket:
                self.put(key, value)

Open addressing: All elements are stored in the array itself. When a collision occurs, probe for The next empty slot using a deterministic sequence.

Strategy	Probe Sequence	Clustering
Linear probing	$h(k), h(k)+1, h(k)+2, \ldots$	Primary clustering
Quadratic probing	$h(k), h(k)+1^2, h(k)+2^2, \ldots$	Secondary clustering
Double hashing	$h(k), h(k)+h'(k), h(k)+2h'(k), \ldots$	No clustering

Robin Hood hashing: A variant of open addressing where elements with shorter probe distances are Favoured. When inserting, if the new element has a longer probe distance than the existing element, Swap them. This minimises the variance of probe lengths, giving more consistent performance.

Hash Function Design

A good hash function should:

Be deterministic. Same key always produces same hash
Be fast. Hash computation is on the critical path
Distribute uniformly. Avoid clustering
Be avalanche-like. A small change in input produces a large change in output

For integers, a common choice is the MurmurHash finaliser (MurmurHash3 mix):

def murmurhash3_mix(key: int) -> int:
    """MurmurHash3 finaliser for 64-bit integers."""
    key ^= key >> 33
    key = (key * 0xff51afd7ed558ccd) & 0xFFFFFFFFFFFFFFFF
    key ^= key >> 33
    key = (key * 0xc4ceb9fe1a85ec53) & 0xFFFFFFFFFFFFFFFF
    key ^= key >> 33
    return key

Randomisation by default via `PYTHONHASHSEED`). This is a security measure against HashDoS attacks. For persistent hashing (e.g., on-disk hash tables), use `hashlib` or a deterministic hash function.

String Matching

Naive Algorithm

Compare the pattern against every possible position in the text.

def naive_search(text, pattern):
    """
    Naive string matching.
    Time: O(n*m) worst case, O(n+m) best case
    Space: O(1)
    """
    n, m = len(text), len(pattern)
    for i in range(n - m + 1):
        match = True
        for j in range(m):
            if text[i + j] != pattern[j]:
                match = False
                break
        if match:
            return i
    return -1

Rabin-Karp Algorithm

Use hashing to compare the pattern with substrings in $O(1)$ expected time per position.

def rabin_karp(text, pattern, base=256, mod=10**9 + 7):
    """
    Rabin-Karp string matching using rolling hash.
    Time: O(n+m) average, O(nm) worst case
    Space: O(1)
    """
    n, m = len(text), len(pattern)
    if m > n:
        return -1

    pattern_hash = 0
    text_hash = 0
    base_m = 1  # base^(m-1) % mod

    for i in range(m):
        pattern_hash = (pattern_hash * base + ord(pattern[i])) % mod
        text_hash = (text_hash * base + ord(text[i])) % mod
        if i < m - 1:
            base_m = (base_m * base) % mod

    for i in range(n - m + 1):
        if text_hash == pattern_hash:
            # Verify to avoid false positives (hash collision)
            if text[i:i + m] == pattern:
                return i
        if i < n - m:
            # Rolling hash: remove leading char, add trailing char
            text_hash = (
                (text_hash - ord(text[i]) * base_m) * base + ord(text[i + m])
            ) % mod
            text_hash = (text_hash + mod) % mod  # ensure non-negative

    return -1

Complexity: $O(n+m)$ average case, $O(nm)$ worst case (all positions are hash matches requiring Verification). Using two independent hash functions reduces the probability of false positives to Negligible levels.

Knuth-Morris-Pratt (KMP) Algorithm

KMP preprocesses the pattern to build a “failure function” (also called the LPS array — longest Proper prefix which is also suffix) that allows the algorithm to skip redundant comparisons.

def build_lps(pattern):
    """
    Build the Longest Proper Prefix-Suffix array for KMP.
    Time: O(m), Space: O(m)
    """
    m = len(pattern)
    lps = [0] * m
    length = 0  # length of the previous longest prefix suffix
    i = 1
    while i < m:
        if pattern[i] == pattern[length]:
            length += 1
            lps[i] = length
            i += 1
        else:
            if length != 0:
                length = lps[length - 1]
            else:
                lps[i] = 0
                i += 1
    return lps

def kmp_search(text, pattern):
    """
    Knuth-Morris-Pratt string matching.
    Time: O(n + m) worst case
    Space: O(m)
    """
    n, m = len(text), len(pattern)
    if m == 0:
        return 0
    if m > n:
        return -1

    lps = build_lps(pattern)
    i = 0  # index in text
    j = 0  # index in pattern

    while i < n:
        if text[i] == pattern[j]:
            i += 1
            j += 1
            if j == m:
                return i - j  # match found
        else:
            if j != 0:
                j = lps[j - 1]
            else:
                i += 1
    return -1

Complexity: $O(n + m)$ worst case — guaranteed linear time, no hash collisions to worry about. The LPS array ensures the text pointer i never moves backward.

Anagram and Grouping Problems

Valid Anagram

def is_anagram(s, t):
    """
    Check if t is an anagram of s.
    Time: O(n), Space: O(1) — fixed alphabet size
    """
    if len(s) != len(t):
        return False
    counts = [0] * 26  # assuming lowercase English letters
    for c in s:
        counts[ord(c) - ord('a')] += 1
    for c in t:
        counts[ord(c) - ord('a')] -= 1
        if counts[ord(c) - ord('a')] < 0:
            return False
    return True

Group Anagrams

from collections import defaultdict

def group_anagrams(strings):
    """
    Group strings by anagram equivalence.
    Time: O(n * k * log k) where k = max string length
    Space: O(n * k)
    """
    groups = defaultdict(list)
    for s in strings:
        key = ''.join(sorted(s))
        groups[key].append(s)
    return list(groups.values())

An alternative using character counts as keys (better for long strings with small alphabets):

def group_anagrams_count(strings):
    """
    Group anagrams using character count tuples as keys.
    Time: O(n * k) where k = max string length
    Space: O(n * k)
    """
    groups = defaultdict(list)
    for s in strings:
        counts = [0] * 26
        for c in s:
            counts[ord(c) - ord('a')] += 1
        groups[tuple(counts)].append(s)
    return list(groups.values())

Matrix Traversal Patterns

Spiral Traversal

def spiral_order(matrix):
    """
    Traverse an m x n matrix in spiral order.
    Time: O(m * n), Space: O(1) (excluding output)
    """
    if not matrix or not matrix[0]:
        return []
    result = []
    top, bottom = 0, len(matrix) - 1
    left, right = 0, len(matrix[0]) - 1

    while top <= bottom and left <= right:
        # Traverse right
        for col in range(left, right + 1):
            result.append(matrix[top][col])
        top += 1

        # Traverse down
        for row in range(top, bottom + 1):
            result.append(matrix[row][right])
        right -= 1

        if top <= bottom:
            # Traverse left
            for col in range(right, left - 1, -1):
                result.append(matrix[bottom][col])
            bottom -= 1

        if left <= right:
            # Traverse up
            for row in range(bottom, top - 1, -1):
                result.append(matrix[row][left])
            left += 1

    return result

Diagonal Traversal

def diagonal_traverse(matrix):
    """
    Traverse an m x n matrix in diagonal zigzag order.
    Time: O(m * n), Space: O(1) (excluding output)
    """
    if not matrix or not matrix[0]:
        return []
    m, n = len(matrix), len(matrix[0])
    result = []
    row, col = 0, 0
    going_up = True

    while len(result) < m * n:
        if going_up:
            while row >= 0 and col < n:
                result.append(matrix[row][col])
                row -= 1
                col += 1
            row += 1
            if col >= n:
                col -= 1
                row += 1
        else:
            while col >= 0 and row < m:
                result.append(matrix[row][col])
                row += 1
                col -= 1
            col += 1
            if row >= m:
                row -= 1
                col += 1
        going_up = not going_up

    return result

Common Problem Patterns

Dutch National Flag

Partition an array into three sections (e.g., values less than, equal to, and greater than a pivot) In a single pass.

def dutch_national_flag(arr, pivot_idx):
    """
    Three-way partition around arr[pivot_idx].
    After: arr[0..lt-1] < pivot, arr[lt..gt] == pivot, arr[gt+1..n-1] > pivot
    Time: O(n), Space: O(1)
    """
    pivot = arr[pivot_idx]
    lt, gt = 0, len(arr) - 1
    i = 0
    while i <= gt:
        if arr[i] < pivot:
            arr[lt], arr[i] = arr[i], arr[lt]
            lt += 1
            i += 1
        elif arr[i] > pivot:
            arr[gt], arr[i] = arr[i], arr[gt]
            gt -= 1
            # Don't increment i — need to examine the swapped element
        else:
            i += 1
    return lt, gt  # boundaries of the equal region

Merge Sorted Arrays

def merge_sorted_arrays(a, b):
    """
    Merge two sorted arrays into one sorted array.
    Time: O(n + m), Space: O(n + m)
    """
    result = []
    i, j = 0, 0
    while i < len(a) and j < len(b):
        if a[i] <= b[j]:
            result.append(a[i])
            i += 1
        else:
            result.append(b[j])
            j += 1
    result.extend(a[i:])
    result.extend(b[j:])
    return result

Merge Sorted Arrays In-Place

def merge_in_place(a, b):
    """
    Merge sorted array b into sorted array a.
    Assumes a has enough buffer at the end to hold b.
    Time: O(n + m), Space: O(1)
    """
    total = len(a) + len(b) - 2  # subtract buffer count
    write = total
    i = len(a) - 1  # last real element in a
    j = len(b) - 1

    while i >= 0 and j >= 0:
        if a[i] > b[j]:
            a[write] = a[i]
            i -= 1
        else:
            a[write] = b[j]
            j -= 1
        write -= 1

    while j >= 0:
        a[write] = b[j]
        j -= 1
        write -= 1

Majority Element (Boyer-Moore Voting)

def majority_element(arr):
    """
    Find the element that appears more than n/2 times (if it exists).
    Boyer-Moore voting algorithm.
    Time: O(n), Space: O(1)
    """
    candidate = None
    count = 0
    for x in arr:
        if count == 0:
            candidate = x
            count = 1
        elif x == candidate:
            count += 1
        else:
            count -= 1

    # Verify (the algorithm only finds a candidate)
    if arr.count(candidate) > len(arr) // 2:
        return candidate
    return None

Common Pitfalls

1. Off-by-One Errors in Window Problems

Sliding window problems are rife with off-by-one errors. The most common: confusing “exclusive upper Bound” with “inclusive upper bound.” Decide on a convention (Python-style [left, right) is Recommended) and stick to it consistently. The number of elements in the window is always right - leftNever right - left + 1.

2. Integer Overflow in Rolling Hashes

Rabin-Karp’s rolling hash computation involves multiplication and addition that can overflow 64-bit Integers for large inputs. Always use modular arithmetic with a large prime modulus (or use Python’s Arbitrary-precision integers, which do not overflow). For production use, consider using two Independent hash functions to reduce collision probability.

3. Forgetting Edge Cases in Two-Pointer Problems

Empty arrays, single-element arrays, arrays with all identical elements, and arrays where no valid Pair exists are all edge cases that two-pointer solutions must handle. Test your solution with [] [1]``[1, 1, 1, 1]And a case where no solution exists.

4. Treating Strings as Arrays of Bytes

In a world of UTF-8, string indexing does not give you bytes — it gives you code points. Reversing a UTF-8 string by swapping bytes produces invalid UTF-8. Reversing by code points is safe but does not Handle grapheme clusters (e.g., the emoji flags sequence). Use language-appropriate string reversal.

5. Hash Map Key Mutability

If you use a mutable object as a hash map key and then mutate it, the object’s hash changes and you Can no longer find it in the map. In Python, this manifests as a dict key becoming invisible after Mutation. Always use immutable types (tuples, frozensets, strings) as keys, or ensure keys are never Mutated after insertion.

6. Prefix Sum Integer Overflow

For large arrays, prefix sums can overflow 32-bit integers. A sum of $10^6$ elements each up to $10^9$ gives $10^{15}$ Which exceeds 32-bit range. Use 64-bit integers (int in Python is always Arbitrary precision, but in C/C++/Java, use long long/long).

7. Ignoring the Difference Between “At Most k” and “Exactly k”

In sliding window problems, “at most k distinct elements” requires shrinking the window when the Count exceeds $k$ While “exactly k distinct elements” requires maintaining two windows (one for at Most $k$ and one for at most $k-1$ ). Conflating these leads to incorrect solutions.

Summary

This topic covers the core concepts of arrays and strings, including underlying theory, practical implementation, and key applications.

Key concepts include:

Big O notation and complexity analysis
searching algorithms (binary, linear)
sorting algorithms (bubble, merge, quick)
graph algorithms (Dijkstra, BFS, DFS)
dynamic programming

Understanding these concepts thoroughly is essential for both examinations and practical programming, and requires both theoretical knowledge and hands-on practice.

Worked Examples

Worked examples demonstrating the application of key concepts are covered in the detailed sub-pages linked above.

Arrays and Strings

Intuition

Array Fundamentals

Memory Layout and Cache Behaviour

Dynamic Arrays

See Also