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

Which statement about a recursive function without a reachable base case is correct?

  1. AIt keeps calling itself until the call stack overflows
  2. BIt returns 0 automatically
  3. CIt runs in O(1) time
  4. DThe compiler always turns it into a loop that ends safely
2.Algorithms & complexityHard

What is the time complexity of building a binary heap from n unsorted elements with the bottom-up method?

  1. AO(log n)
  2. BO(n)
  3. CO(n²)
  4. DO(n log n)
3.Data structuresMedium

An adjacency matrix for a graph with V vertices uses how much memory?

  1. AO(E)
  2. BO(V²)
  3. CO(V + E)
  4. DO(log V)
4.Data structuresHard

Which statement about heapsort is correct?

  1. AIt needs a binary search tree as extra storage, so it uses O(n) extra memory
  2. BIt is stable and needs O(n) extra memory
  3. CIt runs in O(n) time on any input
  4. DIt sorts in place in O(n log n) worst-case time but is not stable
5.Graph theoryMedium

A tree has 3 vertices of degree 3, 2 vertices of degree 4, 3 vertices of degree 5, and all its other vertices are leaves (degree 1). How many leaves L does it have?

3 × deg 32 × deg 43 × deg 5
L =
6.Graph theoryHard
51561341839ABCDEF

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

|V| = 6|E| = 8
W =

3Solutions

1.Algorithms & complexity

Answer: A · It keeps calling itself until the call stack overflows

Why this is the answer

Without a base case the recursion never stops; in practice memory for stack frames runs out and the program crashes.

Answer

A · It keeps calling itself until the call stack overflows
2.Algorithms & complexity

Answer: B · O(n)

Why this is the answer

Summing sift-down costs over the levels gives a convergent series, so heapify runs in linear time.

Answer

B · O(n)
3.Data structures

Answer: B · O(V²)

Why this is the answer

An adjacency matrix stores V × V entries, efficient for dense graphs; adjacency lists use O(V + E), better for sparse graphs.

Answer

B · O(V²)
4.Data structures

Answer: D · It sorts in place in O(n log n) worst-case time but is not stable

Why this is the answer

Heapsort builds a max-heap and extracts the maximum n times within the same array; swaps over long distances can reorder equal keys.

Answer

D · It sorts in place in O(n log n) worst-case time but is not stable
5.Graph theory

Answer: L = 18

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 + 32 = 2(L + 8) − 2
L = 18

Answer

L = 18

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

6.Graph theory

Answer: W = 31

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 (3): take
DE (4): take
AB (5): take
BD (6): take
BE (9): skip, closes a cycle
CE (13): take

Total weight

W = 3 + 4 + 5 + 6 + 13 = 31

Answer

W = 31

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