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

Which definition of the class P is correct?

  1. AProblems that need exponential time on every computer
  2. BProblems whose solutions can be checked but never found in polynomial time
  3. CDecision problems solvable by a deterministic algorithm in polynomial time
  4. DProblems that can be solved in parallel
3.Data structuresMedium

A trie (prefix tree) is especially efficient for:

  1. AStoring matrices
  2. BFinding shortest paths
  3. CPrefix searches such as autocomplete
  4. DSorting numbers
4.Data structuresHard

Why can elements not simply be deleted from a standard Bloom filter by clearing their bits?

  1. ADeleting would increase the false positive rate to 100%
  2. BBloom filters store the full elements, and these cannot be located again once hashed
  3. CThose bits may also belong to other elements, which would cause false negatives
  4. DBits in a Bloom filter cannot be changed after creation
5.Graph theoryMedium
ABCDEFGEuler circuitEuler path, no circuitneither

Does the connected graph have an Euler circuit, an Euler path that is not a circuit, or neither? Click your answer.

|V| = 7|E| = 10
  1. 1Euler circuit
  2. 2Euler path, no circuit
  3. 3neither
6.Graph theoryHard
1029999915134ABCDEFG

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: A · 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

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

Answer: C · Decision problems solvable by a deterministic algorithm in polynomial time

Why this is the answer

P contains decision problems with a deterministic polynomial-time algorithm, such as checking whether a graph is connected.

Answer

C · Decision problems solvable by a deterministic algorithm in polynomial time
3.Data structures

Answer: C · Prefix searches such as autocomplete

Why this is the answer

Tries store strings by characters along paths, so lookups and prefix queries take time proportional to the key length.

Answer

C · Prefix searches such as autocomplete
4.Data structures

Answer: C · Those bits may also belong to other elements, which would cause false negatives

Why this is the answer

Clearing a shared bit would make another element look absent; counting Bloom filters keep counters instead of bits to allow deletion.

Answer

C · Those bits may also belong to other elements, which would cause false negatives
5.Graph theory

Answer: Euler path, no circuit

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.

Degrees

deg(A) = 2, deg(B) = 2, deg(C) = 3, deg(D) = 4, deg(E) = 4, deg(F) = 2, deg(G) = 3

Odd vertices

C, G → 2
Euler path, no circuit

Answer

Euler path, no circuit

The Euler path starts at one odd vertex and ends at the other.

6.Graph theory

Answer: d(F) = 28

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): B: 10
B (10): C: 12, D: 19, G: 19
C (12)
D (19): E: 28, F: 28
G (19): E: 23
E (23)
F (28)

Shortest path

A → B → D → F
d(F) = 28

Answer

d(F) = 28

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