algorithm

relation to e_1..e_{r−1}: NO relation in general (c_j elsewhere)! BUT: e_r alive during [j0, r−1] requires connectors-at-j = path(c_j→e_r) alive-at-j: those connectors: edges on tree-path from PAST targets to e_r: THEY WERE THE EDGES VIA WHICH e_r's aliveness was maintained in the ACTUAL H-evolution... CLAIM: (b) automatically satisfiable ⇔ for each j: |Steiner(c_j ∪ committed-at-j ∪ {e_r})| ≤ m−1 AND the connector edges are themselves "alive-runs" — since connectors ⊆ swept-history and their own gaps... I'M GOING TO TRUST-AND-VERIFY: implement the greedy with rule [prefix + range_max(used) check + range-increment on save + lastTouch updates + used[i] init 0], WITHOUT explicit (b), and see if it matches brute force. If mismatches appear, study them and refine (probably (b) shows up as needed in branchy cases — then find its efficient form: likely: "e_r's window must also satisfy: lastTouch(e_r) ≥ aliveSinceChain..." something like clamping j0 by the previous edge's savedWindow: windowStart(e_r) = max(lastTouch(e_r), windowStart(e_{r−1}))?? — because e_r's aliveness before e_{r−1}-chain existed can't connect?? hmm no — connectivity at past leg j goes through PAST paths not the current one. GAAAH. Data first!)

Also don't forget: what about used[] at leg i itself (current): saved edges e_1..e_t: they're alive NOW: they'll be part of H_i: commit their coverage of leg i when... their windows [j0, i−1] end at i−1; their CURRENT aliveness [i, nextTouch) = committed at NEXT save. But capacity AT leg i: |H_i| ≤ m−1: H_i ⊇ saved-prefix + (other still-alive stuff): future saves with windows ∋ i will range-check used[i]: used[i] must count... committed-covering-i = edges saved LATER with window ∋ i ✓ retro ✓ consistent: used[i] starts 0 ✓ ✓ and the just-saved prefix's coverage of leg i gets counted retroactively if/when they're saved AGAIN later ✓ elegant!!

AND the S_0 edges: lastTouch = 0; used[0] = ?: leg-0 aliveSet = S_0 exactly = m−1 edges: FULL: used[0] = m−1?? Then any save with window ∋ 0 fails range check... but S_0 edges SHOULD be saveable (a1!!): e = (v0,a1): window [lastTouch=0, i−1] ∋ 0: used[0] = m−1 = FULL ⇒ blocked?! WRONG. Fix: e ∈ S_0's own presence at leg 0 is FREE (it's already there!) — the range check should be [j0 + 1?, i−1]?? window semantic: e alive at ENDS of legs j0..i−1; at leg j0 e is present ANYWAY at the moment of its touch — is end-of-leg-j0 aliveness

mathalgorithmreasoning tracecompetitive programming

mid-leg ⊆?? — the physical blob: connected ALWAYS. My whole leg-granular abstraction glossed this; the convexity argument: |Steiner(head ∪ R)| with FIXED R: ≤ max(ends) ✓ so IF R fixed during leg (spanning commitments) the mid-leg Steiner (incl. connectors!) ≤ max of end-Steiners: end-Steiner = |Steiner(c_j ∪ R)| ≤ hmm is end-Steiner = used[j]?? used[j] counts committed edges; Steiner(c_j ∪ committed) ⊇ committed + connectors: CONNECTORS AT END-OF-LEG: must ⊆ committed (else uncounted occupancy!!) — condition (b) again!! — (b) ensures end-Steiners = committed exactly ⇒ mid-leg (fixed-R part) ≤ max(used[j−1], used[j]) + path... NO the convexity: Steiner(head ∪ R) as head walks c_{j−1} → c_j: = |Steiner(R)| + dist(head, Steiner(R)): max at ends: ≤ max(used[j−1], used[j]) ✓ THEN + the leg's own touched-kept edges (B(r)) + x: total ≤ max(used[j−1], used[j]) + B + 1: hmm B ⊆ used[j]-parts... I'M DROWNING — EMPIRICS!!! Let me define v1 conservatively ("reserve 1": check ≤ m−2 means post-commit used ≤ m−1?? no wait — hmm, reserve-1 = require used[j] + 1(new) ≤ m−2?? ugh. Let me parametrize: CHECK: max(used[window]) + take ≤ CAP with CAP ∈ {m−1, m−2} and compare both against brute. ALSO test the (b)-connector-check variant. DATA DATA DATA. GO.

(Also, sanity: m−1 = capacity for edges; when m=2: CAP = m−2 = 0 ⇒ NO saves ever with reserve-rule — but m=2 analysis showed saves = immediate-backtrack (the returning edge saved: window [i−1, i−1] hmm: e touched leg i−1 (as the LAST edge into c_{i−1}) and leg i (first edge out): window [i−1, i−1]: used[i−1] must ≤ m−2 = 0: if no other commitments: 0 ≤ 0 ✓ take 1: used[i−1] = 1 = m−1: mid-leg-i−1 conflicts: leg i−1's OTHER crossings before e: they saw e?? e crossed LAST in leg i−1: earlier crossings: e not yet active ✓ fine; and spanning commitments at i−1 = 0 ✓ feasible physically: blob = the edge sliding, at c_{i−1} blob = last edge = e, then leg i starts crossing e (in cache!) SAVED ✓✓ so CAP = m−2 as "max(used) + take ≤ m−1" i.e. check max(used[window]) ≤ m−1 − take... for take = 1: used ≤ m−2 ✓ matches (i). Good — so v1: room = (m−1) − maxUsed; take = min(len, room) — with the caveat the mid-leg overflow concern suggests maybe room = (m−2) − maxUsed sometimes. m=2: room = 1 − used: used = 0 → take 1 ✓ good sign for room = m−1 − maxUsed.)

Let me code and find out. After exploration, write final sol.cpp + a slow-reference cpp for

mathalgorithmreasoning tracecompetitive programming

window [τ, i-1] must satisfy... the constraint isn't uniform-per-leg: legs j where all uncommitted-or-later-crossing happens mid-leg" effectively have capacity m−2 for SPANNING commitments?! Define: commitments spanning INTO leg j from before (window ⊇ [j−1, j]) vs starting at j: the mid-leg-crossing coexists with: spanning ones (all active) + started-at-j ones touched earlier + x. Hmm — cleanest fix: **count capacity on HALF-LEG granularity or on "crossing-slots": treat constraint as: for every leg j: (#committed spanning j−1→j) + (#committed starting at j) + (1 if leg j has ≥1 uncommitted crossing...) ≤ m−1 — WAIT simpler: EVERY leg j with d_j ≥ 1 crossings: at the FINAL crossing x_last of leg j: active ≥ (all committed ∋ j EXCEPT... committed-starting-at-j-with-touch-AFTER... none after the last) so = used[j] + [x_last uncommitted]: ⇒ constraint: used[j] + [lastEdge(leg j) not committed-∋-j] ≤ m−1. And EARLIER crossings: active = spanning + started-touched-before + 1: ≤ spanning + (started total) + 1 = used[j] + 1 hmm same bound unless x itself started: ⇒ uniform SAFE constraint: **used[j] ≤ m−2 whenever leg j has any crossing "not committed at j"... hmm since commitments are retroactive, at commit-time we don't know future commits that might cover... GRRR. RESOLUTION: charge the current-leg's OWN saved-prefix occupancy EAGERLY: when leg i saves e_1..e_t: ALSO commit their occupancy AT LEG i (they're likely re-saved later extending, but their [i, ·] would then start at i: DOUBLE-COUNT at leg i!! unless later windows = [i+1?? no their lastTouch = i so window [i, ·] ✗ overlap... FIX: commit window [τ, i] (inclusive!) at save; next save of e at i' has window [lastTouch = i, i'] → overlap at leg i... adjust: next window [i+1?? — Let me re-derive cleanly what the occupancy of a saved edge is: e saved at leg i (touched legs τ then i): e alive [touch@τ .. touch@i] continuously — end-of-leg occupancies: legs τ, τ+1, ..., i−1 (end-of-leg-i: e alive AT its touch mid-leg-i; till end-of-i only if kept FURTHER — undecided). So window [τ, i−1] ✓ as before, AND ADDITIONALLY: e's activity DURING legs τ and i (partial mid-leg): the mid-leg conflicts: (1) during leg τ after e's touch: e coexists with leg-τ's later crossings: e occupies "the tail of leg τ" — captured by used[τ] end-count ✓ BUT the mid-leg-τ crossings AFTER e: they see e active: count: (used[τ]-committed-spanning + e + ...) ≤ hmm THE CLEAN FIX: constraint per leg j: used[j] ≤ m−2 (strictly reserve one slot for "the current crossing") — i.e., commitments never fill to m−1;

mathalgorithmreasoning tracecompetitive programming

own touched-kept edges (B(r)) + x: total ≤ n
parts... I'M DROWNING — EMPIRICS!!! Let
m−2 means post-commit used ≤ m−1?? no
w) ≤ m−2?? ugh. Let me parametrize: CHE
? ∈ {m−1, m−2} and compare both against
ant. DATA DATA DATA. GO.

mathalgorithmreasoning tracecompetitive programming

committed only ✓✓. BUT — mid-leg capacity: while walking leg j, the head carries... the active set mid-leg = committed-spanning + current-edge + connectors-to-committed: |...| ≤ m−1 MID-LEG?? — the current edge + committed: committed-spanning-legs (j−1, j) hmm: committed windows cover leg-ENDS; MID-leg-j active = (committed with window ∋ j−1 AND ∋ j?? those alive across (j−1→j) = window ⊇ {j−1, j}... an edge with window ending exactly j−1: alive at end-of-(j−1), droppable during leg j ✓; window starting at j: alive from... its touch MID-leg j: before its touch: not active ✓. So mid-leg-j active ⊆ {committed ∋ both j−1,j} ∪ {committed ∋ j−1 not yet dropped: droppable ✓ drop them} ∪ {committed ∋ j: activate as touched... their touch = ON leg j?? window starting j with touch at leg j: e touched at leg j and saved LATER with window [j, ·]: occupies from its touch onward ✓} ∪ current-edge ∪ connectors: WORST mid-leg count = |{∋ j−1 ∩ ∋ j}| + |{∋ j, touched-so-far}| + 1 + connectors... could this EXCEED m−1 while both end-checks pass??? The end-of-leg-j count = used[j] + (uncommitted) ≤ m−1 checked; mid-leg: {∋ j−1 ∩ ∋ j} ∪ {∋ j touched-so-far} ⊆ {∋ j} hmm NO: {∋ j−1 ∩ ∋ j} ⊆ {∋ j} ✓ and so mid-leg committed-active ⊆ committed ∋ j ∪ {current edge} ⇒ count ≤ used[j] + 1 ≤ (m−2) + 1 = m − 1 ✓ PHEW — wait used[j] when fully loaded = m−1 (after commits): then mid-leg +1 current-edge = m: VIOLATION?! Hmm: used[j] = m−1 means m−1 edges committed-alive at end-of-leg-j; mid-leg while walking current edge x (a miss, new) + the m−1 committed: do all m−1 committed NEED activity at the mid-leg moment?? Committed ∋ j = alive at END of leg j; mid-leg they... an edge committed [a, b] with a < j ≤ b: alive THROUGH leg j: yes active mid-leg ✓; an edge committed [j, b] (touch at leg j): active from its touch: if touched BEFORE x: active ✓. So count = |{committed [a≤j−1, ≥j]}| + |{committed [j, ·] touched before x}| + 1(x) + connectors(⊆ committed ✓ counted). If all used[j] = m−1 are the first two kinds before x's touch: m−1 + 1 = m ✗✗ EXCEEDS!!! IS THIS REACHABLE?: x = a later-position edge on leg j's path, and m−1 commitments already covering leg j at x's crossing-time... e.g., committed-set = stuff near c_{j−1} retained-to-future (window ∋ j) + leg-j's early edges saved-later-with-window-from-j: while crossing x mid-leg: active = those + x: m ⇒ contradiction ⇒ my leg-granular capacity is TOO LOOSE — real constraint binds MID-LEG ⇒ v1 might OVERCOUNT savings vs brute!! — earlier I "proved" end-checks suffice via dist-convexity:

mathalgorithmreasoning tracecompetitive programming