Data Structures and Algorithms: cram pack

Formula sheets, the key method steps for every topic and 6 practice problems, with full solutions at the back.

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1Formulas and methods

Graph theory

Handshake lemma
Σ deg(v) = 2|E|
Complete graphs
|E(Kn)| = n(n − 1)/2, |E(Km,n)| = m·n
Regular graph
|E| = n·k/2
Tree and forest
|E| = n − 1, |E| = n − c
Leaves of a tree
L = 2 + Σ (deg(v) − 2) over deg(v) ≥ 3
Euler
circuit: all degrees even; path: exactly 2 odd degrees
Hamilton circuits in Kn
(n − 1)!/2
Dijkstra update
d(v) = min(d(v), d(u) + w(u, v))
Walks of length L
(Mᴸ)ij

Method

  1. Write down the degree of every vertex; the degrees add up to twice the number of edges.
  2. For Euler questions, check that the graph is connected and count the vertices with odd degree.
  3. For trees, use n − 1 edges and the handshake lemma.
  4. For Dijkstra, keep a table of tentative distances and settle the smallest unsettled vertex each round.
  5. For a minimum spanning tree, sort the edges by weight and skip every edge that closes a cycle (Kruskal), or grow one tree from a start vertex (Prim).

2Practice problems

1.Algorithms & complexityMedium

Is it correct to say that an O(n²) algorithm is also O(n³)?

  1. AYes, but only if n is smaller than 100
  2. BNo, because n³ is smaller than n² for large n
  3. CNo, because Big O must always state the exact growth rate of the function
  4. DYes, because Big O is an upper bound and n² grows no faster than n³
2.Algorithms & complexityHard

Why does any comparison-based search in a sorted array need Ω(log n) comparisons in the worst case?

  1. AA decision tree with n possible outcomes must have depth at least log₂ n
  2. BComparisons take logarithmic time each
  3. CBinary search is the only possible algorithm, and it needs log₂ n comparisons
  4. DEvery element must be compared at least once
3.Data structuresMedium

How does extract-min work in a binary min-heap?

  1. ARemove the root and shift all elements one place left
  2. BMove the last element to the root and sift it down
  3. CSearch all leaves for the new minimum
  4. DRemove the last element of the array
4.Data structuresHard

Which structure lets you compute the maximum of every window of size k in an array in O(n) total?

  1. AA singly linked list of the window's elements
  2. BA min-heap rebuilt for every window
  3. CA deque holding indices in decreasing order of value
  4. DA sorted array updated by shifting
5.Graph theoryMedium

A tree has 3 vertices of degree 4, 3 vertices of degree 2, and all its other vertices are leaves (degree 1). How many leaves L does it have?

3 × deg 43 × deg 2
L =
6.Graph theoryHard
1912613111618210ABCDEF

Find the total weight W of a minimum spanning tree of the weighted graph.

|V| = 6|E| = 9
W =

3Solutions

1.Algorithms & complexity

Answer: D · Yes, because Big O is an upper bound and n² grows no faster than n³

Why this is the answer

Big O is an upper bound, so the statement is true but not tight; Θ(n²) states the exact growth rate.

Answer

D · Yes, because Big O is an upper bound and n² grows no faster than n³
2.Algorithms & complexity

Answer: A · A decision tree with n possible outcomes must have depth at least log₂ n

Why this is the answer

Each comparison at most halves the set of remaining possibilities, so distinguishing n positions needs at least log₂ n comparisons.

Answer

A · A decision tree with n possible outcomes must have depth at least log₂ n
3.Data structures

Answer: B · Move the last element to the root and sift it down

Why this is the answer

The last element fills the hole at the root and is swapped with its smaller child until the heap property holds, taking O(log n).

Answer

B · Move the last element to the root and sift it down
4.Data structures

Answer: C · A deque holding indices in decreasing order of value

Why this is the answer

The front of the deque is always the window's maximum; each index enters and leaves the deque once, so the work is linear.

Answer

C · A deque holding indices in decreasing order of value
5.Graph theory

Answer: L = 8

Euler, Hamilton and trees: a connected graph has an Euler circuit (every edge once, back to the start) when every vertex has even degree, and an Euler path that is not a circuit when exactly two vertices have odd degree; the path then runs from one odd vertex to the other. A Hamilton circuit visits every vertex once; Kn has (n − 1)!/2 of them. A tree on n vertices has n − 1 edges, and a connected graph with m edges keeps a spanning tree after removing m − n + 1 edges.

Handshake lemma for a tree

L + 18 = 2(L + 6) − 2
L = 8

Answer

L = 8

Shortcut: L = 2 + Σ (d − 2) over the vertices with degree d ≥ 3.

6.Graph theory

Answer: W = 43

Shortest paths and spanning trees: Dijkstra’s algorithm settles one vertex per round: the unsettled vertex with the smallest tentative distance. Its distance is then final, and every neighbour v gets d(v) = min(d(v), d(u) + w(u, v)). Kruskal’s algorithm builds a minimum spanning tree by taking the edges from light to heavy and skipping any edge that would close a cycle; Prim’s algorithm grows one tree from a start vertex by always adding the lightest edge that leaves it. A spanning tree of n vertices has n − 1 edges.

Kruskal’s algorithm

EF (2): take
BC (6): take
BE (10): take
CE (11): skip, closes a cycle
AC (12): take
BD (13): take

Total weight

W = 2 + 6 + 10 + 12 + 13 = 43

Answer

W = 43

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