Codeforces VP/琛ラ灏忚 (鎸佺画濉潙)

Posted 2022-10-07

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Codeforces VP/琛ラ灏忚

1149 C. Tree Generator

缁欎綘涓€妫垫爲鐨勬嫭鍙峰簭鍒楋紝姣忔浜ゆ崲涓や釜鎷彿锛岀淮鎶ゆ瘡娆′氦鎹箣鍚庣殑鐩村緞銆?/p>

? 鑰冭檻鎷彿搴忓垪缁存姢鏍戠殑璺緞淇℃伅鍜岋紝鏄皢宸︽嫭鍙风湅鍋?\(-1\) 锛屽彸鎷彿鐪嬪仛 \(1\) 锛岄偅涔堜竴娈电珫鐩村悜涓婄殑璺緞鍙互琛ㄧず涓烘嫭鍙峰簭鍒楃殑涓€涓尯闂村拰锛屼竴娈电珫鐩村悜涓嬬殑璺緞鍙互鐪嬪仛鎷彿搴忓垪鐨勪竴涓尯闂村拰鐨勭浉鍙嶆暟銆傛垜浠缁存姢鐨勬槸鏍戠殑鐩村緞锛屼篃灏辨槸涓€娈佃繛缁殑鍜屽噺鍘荤揣闅忓叾鍚庣殑涓€娈佃繛缁殑宸€傚叿浣撴潵璇村氨鏄?br /> \[ \max_\forall [l,r]\\sum_i=l^kw[i]-\sum_i=k+1^r w[i]\ \]
? 鍙互绠€鍗曠殑璇佹槑濡傛灉閫夊彇鐨勫尯闂翠笉鑳戒唬琛ㄦ爲涓婄殑璺緞锛屽彇鍒扮殑寮忓瓙鐨勫拰涓€瀹氫笉鏄渶浼樿В锛岀嚎娈垫爲缁存姢杩欎釜寮忓瓙鍗冲彲銆傜淮鎶よ捣鏉ョ暐鏈夐夯鐑︼紝绫讳技浜庣淮鎶ゅ尯闂存渶澶у瓙娈靛拰锛屾洿鏂扮殑鏃跺€欓渶瑕佸璁板嚑涓俊鎭€傚鏉傚害 \(\mathcal O (n\log n)\) 銆?/p>

1149 D. Abandoning Roads

\(n\) 涓偣 \(m\) 鏉¤竟涓斾粎鏈変袱绉嶈竟鏉?\(a, b (a<b)\) 鐨勬棤鍚戠畝鍗曞浘 \(G\)锛?span class="math inline">\(\forall i\) 姹傚嚭 \(G\) 鐨勬墍鏈夋渶灏忕敓鎴愭爲涓紝\(1-i\) 鐨勮矾寰勭殑鏈€灏忛暱搴︺€?span class="math inline">\(1\leq n\leq 70,n-1\leq m \leq 200\) 銆?/p>

? 杩欎釜棰樻垜涓€寮€濮嬫兂浜嗕竴涓敊璇殑鍋氭硶锛屽厛姹傚嚭鏈€灏忕敓鎴愭爲涓?\(b\) 杈圭殑鏁伴噺璁颁负 \(k\) 锛岀劧鍚庡垎鍒眰鍑?\(1\) 鍒板悇涓偣缁忚繃涓嶈秴杩?\(k\) 鏉?\(b\) 杈圭殑鏈€鐭矾銆傝繖涓仛娉曠殑閿欒涔嬪鍦ㄤ簬锛岃櫧鐒跺彲浠ヨ瘉鏄庤捣濮嬬鐐归兘鍦ㄨ矾寰勪笂鐨?\(a\) 杈逛笉浼氱幆鍒囨帀璺緞涓婄殑 \(b\) 杈癸紝浣嗘槸鍙浠ゅ绔偣鍦ㄨ矾寰勪笂灏辫兘鐜垏浜嗐€?/p>

? 鏈熼棿鎴戣繕鑰冭檻浜嗗彟澶栦竴绉嶅仛娉曪紝鍚庢潵寰楃煡鍚庝竴绉嶆€濊矾鏄彲浠ユ墿灞曞埌姝ｈВ鐨勩€傝€冭檻灏嗘墍鏈?\(a\) 杈圭缉鎴愯嫢骞蹭釜鑱旈€氬潡锛屼笉闅捐瘉鏄庯紝濡傛灉涓€鏉＄敓鎴愭爲鐨勮矾寰勭寮€浜嗕竴涓仈閫氬潡鍚庡張杩涘叆浜嗚繖涓仈閫氬潡锛岃繖妫电敓鎴愭爲涓€瀹氫笉鏄渶灏忕敓鎴愭爲锛屽洜涓哄叾閫夋嫨浜嗕細琚幆鍒囩殑 \(b\) 杈广€傛鏃剁洿鎺ュぇ鍔涚姸鍘?\(dp\) 鐨勫鏉傚害鏄?\(O(2^nm)\) 鐨勩€?/p>

? 鍐嶅線涓嬪垎鏋愶紝濡傛灉涓€涓仈閫氬潡澶у皬 \(\leq 3\) 锛岄偅涔堝嵆浣垮瓨鍦ㄧ寮€杩欎釜鑱旈€氬潡鍚庡張鏉ュ埌杩欎釜鑱旈€氬潡鐨勮矾寰勶紝杩欐潯璺緞涔熷繀鐒朵笉鏄渶鐭矾锛屽洜涓鸿繖鏍疯仈閫氬潡鍐呴儴鏈€闀跨殑杈规槸 \(2a\) 锛岃€屽仛涓婅堪鎿嶄綔闇€瑕?\(2b\) 銆傛墍浠ュ彲浠ョ洿鎺ュ拷瑙嗘帀鎵€浠ュぇ灏?\(\leq 3\) 鐨勮仈閫氬潡鐨勯檺鍒讹紝澶嶆潅搴︿紭鍖栧埌 \(\mathcal O (2^\fracn4m)\)銆?/p>

1168 C. And Reachability

鏈変竴涓暱搴︿负 \(n\) 鐨勫簭鍒?\(A\) , \(i,j\) 涔嬮棿鏈変竴鏉℃湁鍚戣竟褰撲笖浠呭綋 \(i < j, A[i] \text and A[j] > 0\) 锛屾瘡娆＄粰鍑?\(x, y(x<y)\) 锛岃闂粠 \(x\) 鍑哄彂鏄惁鑳藉埌杈?\(y\) 銆?/p>

? 鎸変綅鑰冭檻锛屽浜庡綋鍓嶆暟 \(A[x]\) 锛屽叾鑳界洿鎺ュ埌杈惧悗闈㈡墍鏈夊拰鍏舵湁鍏卞悓浣嶇殑鏁般€傛樉鐒讹紝瀵逛簬姣忎竴浣嶏紝杈冨墠闈㈢殑鏁拌兘鍒拌揪杈冨悗闈㈢殑鏁帮紝鎵€浠ュ彧闇€瑕佺淮鎶ゆ瘡涓€涓暟 \(A[x]\) 鍏惰兘鍒拌揪鏌愪竴浣嶆湁 \(1\) 鐨勬渶鍓嶉潰鐨勬暟鐨勪綅缃嵆鍙€傛渶鍚庡垽鏂?\(A[y]\) 鐨勬煇涓€浣?\(1\) 鏄惁鑳借 \(A[x]\) 鍦ㄥ叾涔嬪墠鍒拌揪銆傚鏉傚害 \(\mathcal O(n\log^2 n)\) 銆?/p>

1168 D. Anagram Paths

鏈変竴妫典簩鍙夋爲锛屾瘡鏉¤竟涓婃湁灏忓啓瀛楁瘝鎴栬€?\('?'\) 锛屾瘡娆′慨鏀逛竴鏉¤竟涓婄殑瀛楃锛岃姹傜淮鎶ゆ墍鏈夊彾瀛愬埌鏍圭殑璺緞涓婄殑 \('?'\) 濉负浠绘剰灏忓啓瀛楁瘝鍚庢槸鍚﹁兘閲嶆帓杈惧埌涓€鑷达紝 濡傛灉鍙互锛岃繕闇€杈撳嚭姣忕瀛楃鐨勬渶澶у彲鑳藉嚭鐜版鏁般€?/p>

? 棣栧厛濡傛灉鎵€鏈夊彾瀛愭繁搴︿笉鐩哥瓑锛屾棤璁烘€庝箞鏇存敼杈逛笂鐨勫瓧绗﹂兘鏄棤瑙ｃ€?/p>

? 鐒跺悗浠?\(f[x][c]\) 琛ㄧず鎵€鏈?\(x\) 瀛愭爲鍐呯殑鍙跺瓙鍒?\(x\) 鐨勮矾寰勪笂瀛楃 \(c\) 鐨勬暟閲忔渶澶у€硷紝\(len[x]\) 琛ㄧず \(x\) 鍒板彾瀛愮殑璺濈銆?/p>

鎴戜滑鍙互褰掔撼璇佹槑锛岃兘澶熼噸鎺掕揪鍒颁竴鑷村綋涓斾粎褰?/p>

?
\[ \forall x \sum_c f[x][c]\leq len[x] \]
? 鑰冭檻褰?\(x\) 鏄彾瀛愭椂鍊欙紝鏄剧劧鍏呰銆傚浜庝换鎰忛潪鍙跺瓙鑺傜偣 \(x\)锛屽亣璁惧叾瀛愭爲鍐呮墍鏈夌偣閮芥弧瓒虫潯浠讹紝姝ゆ椂蹇呰鎬ф樉鐒躲€傚厖鍒嗘€ц€冭檻鍏堝皢鍘熷厛鐨勬墍鏈夎矾寰勬帓濂斤紝瀵逛簬鏂板姞杩涙潵鐨勮竟锛屽鏋滄槸涓€鏉★紝鍒欏繀鐒跺悎娉曘€傚鏋滄槸涓ゆ潯杈癸紝瑕佹弧瓒冲紡瀛愬繀椤绘敼鍙樼殑 \(f[x][c]\) 鏁伴噺 \(<2\) 锛屾鏃跺彟涓€杈逛細鏈夊鍑烘潵鐨勯棶鍙峰～涓婅繖涓瓧绗︺€?/p>

? 鐩存帴鏆村姏缁存姢 \(f[x][c]\) 澶嶆潅搴︽槸 \(O(nq)\) 鐨勶紝浣嗘槸涔嬪墠鐨勮瘉鏄庡惎鍙戞垜浠紝瀵逛簬鍙湁涓€涓瀛愮殑鑺傜偣锛屽彲浠ョ洿鎺ュ皢璇ヨ妭鐐逛笌瀛╁瓙鍚堝苟涓嶅奖鍝嶇瓟妗堛€傜劧鍚庤繖閲屾湁涓€涓簩鍙夋爲鐨勫皬濂楄矾锛屽鏋滆繖鏍峰悎骞讹紝浜屽弶鏍戠殑娣卞害涓嶈秴杩?\(\mathcal O(\sqrt n)\) 銆?/p>

? 璇佹槑锛屽鏋滀簩鍙夋爲涓嶅瓨鍦ㄥ彧鏈変竴涓効瀛愮殑鑺傜偣锛岄偅涔堢 \(i\) 灞傜殑娣卞害涓€瀹氳澶т簬绗?\(i-1\) 灞傜殑娣卞害锛屾鏃舵繁搴︽渶澶氫负 \(\mathcal O(\sqrt n)\) 锛岃繖鏍风洿鎺ユ毚鍔涗慨鏀?\(dp\) 鍊煎嵆鍙紝澶嶆潅搴?\(\mathcal O(q\sqrt n)\)銆傚綋鐒剁洿鎺ュ姩鎬?\(dp\) 涔熸槸鍙互鐨勩€?/p>

1188 D. Make Equal

鏈変竴涓簭鍒?\(a\) 锛屾瘡娆″彲浠ヨ姳璐?\(1\) 鐨勪唬浠峰皢鍏朵腑涓€涓暟鍔犱笂 \(2\) 鐨勫箓娆★紝姹傚皢鎵€鏈夋暟鍙樻垚鐩哥瓑鐨勬暟鏃舵墍闇€鐨勬渶灏戜唬浠枫€?/p>

? 鍏堝皢 \(a\) 浠庡皬鍒板ぇ鎺掑簭锛岃嫢鎴戜滑灏?\(a_n\) 鏈€缁堝姞涓婁簡 \(S\) 锛屽垯 \(a_i\) 鏈€缁堝姞涓婁簡 \(S+a_n-a_i\) 銆傞偅涔堥棶棰樿浆鍖栦负閫夋嫨涓€涓暟 \(S\) 锛屾渶灏忓寲 \(\sum_i=1^n bit(S+a_n-a_i)\) 銆?/p>

? 鑰冭檻鎸変綅 \(dp\) 鍐崇瓥 \(S\) 鐨勬瘡涓€浣嶉€?\(0\) 杩樻槸閫?\(1\) 锛屽洜涓烘渶缁堣澶勭悊鐨勮础鐚槸绛変簬 \(a_n-a_i, S\) 杩欎竴浣嶆暟鐩稿姞浠ュ強涓婁竴浣嶇殑杩涗綅鍐冲畾鐨勶紝鎴戜滑杩橀渶瑕佽褰曞摢浜涙暟鍦ㄥ綋鍓嶇姸鎬佷笅琚繘浣嶄簡銆傚彲浠ュ彂鐜帮紝鍋囪鑰冭檻鍒颁簡鍓?\(k\) 浣嶏紝褰撳墠浣嶄細杩涗綅鐨勬暟涓€瀹氭槸鎸夌収 \(\mod 2^k\) 鎺掑簭缁撴灉涓嬬殑涓€涓悗缂€锛?span class="math inline">\(dp\) 鐨勬椂鍊欒褰曚竴涓嬭繖涓悗缂€鐨勯暱搴﹀嵆鍙紝澶嶆潅搴?\(\mathcal O (n\log n)\) 銆?/p>