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

You sort an array of n elements once and then answer k membership queries with binary search. What is the total cost?

  1. AO(n + k)
  2. BO(nk)
  3. CO(n log n + k log n)
  4. DO(k n log n)
2.Algorithms & complexityHard

Karatsuba multiplication satisfies T(n) = 3T(n/2) + O(n). What does this solve to?

  1. AΘ(n log n)
  2. BΘ(nlog₂ 3), about n1.585
  3. CΘ(n²)
  4. DΘ(n3)
3.Data structuresMedium

Why can a binary heap be stored in an array without any pointers?

  1. AHeaps never contain more than 100 elements
  2. BIt is a complete binary tree, so positions follow simple index formulas
  3. CAll keys in a heap are equal
  4. DArrays automatically keep the keys in sorted order, so no pointers are needed
4.Data structuresHard

Which statement about heapsort is correct?

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

A connected graph has 16 vertices and 25 edges. How many edges k must be removed to leave a spanning tree?

|V| = 16|E| = 25
k =
6.Graph theoryHard
1032142913118615ABCDEFG

Use Dijkstra’s algorithm to find the length d(F) of the shortest path from A to F in the weighted graph.

start: Aend: F
d(F) =

3Solutions

1.Algorithms & complexity

Answer: C · O(n log n + k log n)

Why this is the answer

Sorting costs O(n log n) and each query O(log n); this beats k linear scans (O(nk)) when k is large.

Answer

C · O(n log n + k log n)
2.Algorithms & complexity

Answer: B · Θ(nlog₂ 3), about n1.585

Why this is the answer

Since log₂ 3 ≈ 1.585 > 1, the leaves dominate by the master theorem, beating the O(n²) schoolbook method.

Answer

B · Θ(nlog₂ 3), about n1.585
3.Data structures

Answer: B · It is a complete binary tree, so positions follow simple index formulas

Why this is the answer

Completeness means there are no gaps, so the children of index i are always at 2i + 1 and 2i + 2.

Answer

B · It is a complete binary tree, so positions follow simple index formulas
4.Data structures

Answer: B · 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

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

Answer: k = 10

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.

Edges of a spanning tree

16 − 1 = 15

Edges to remove

k = 25 − 15 = 10

Answer

k = 10
6.Graph theory

Answer: d(F) = 24

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.

Dijkstra round by round

Each line: the vertex settled, its final distance, and the neighbours whose distance improves.

A (0): C: 10
C (10): B: 13, E: 24, G: 18
B (13): D: 15
D (15): E: 17, F: 24
E (17)
G (18)
F (24)

Shortest path

A → C → B → D → F
d(F) = 24

Answer

d(F) = 24

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